Working with Distributions (CIE A Level Maths: Probability & Statistics 1)

State the conditions that must be satisfied to be able to model a random variable  with a binomial distribution .

A fair spinner has 8 sectors labelled with the numbers 1 through 8. For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

The random variable  is the number of the sector that the spinner lands on when it is spun.

The random variable  is the number of times the spinner is spun until it lands on ‘7’ for the first time.

The random variable  is the number of times the spinner lands on a prime number when it is spun twelve times.

On the first spin, it is a ‘win’ if the spinner lands on an even number. On subsequent spins it is a ‘win’ if the spinner lands either on the same number as the previous spin or on a factor of the number from the previous spin.  The random variable    is the number of wins when the spinner is spun ten times.

For the random variable defined in (b)(ii) above, give the name of the probability distribution that would be appropriate for modelling .

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

A student cuts a one-metre length of rope into two pieces at a random point. The random variable is the length of the shorter of these two pieces.

You ask a sample of students in your school about their preferences for after-school activities. The random variable is the number of students who say they prefer participating in lawn bowling.

People are chosen at random from the UK population. The random variable  is the age of a randomly selected person, defined as the age which they turned on their most recent birthday.

Each of the following histograms shows the distribution of results when a large number of measurements of the specified random variables –,  or  – are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable.

For each of the following binomial random variables, :

• state, with reasons, whether  can be approximated by a normal distribution
• if appropriate, write down the normal approximation to  in the form  , giving the values of  and .

The random variable 

Explain why a normal distribution can be used to approximate .

Find, using the normal approximation:

Using the normal approximation, find the largest value of  such that

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution

On a casino roulette wheel, the probability of the ball landing on a black number is   .

The wheel is spun 30 times, and the ball lands on a black number  times.

Find

In a separate experiment, the wheel is spun 1000 times and , the number of times the ball lands on a black number, is recorded.

Write down the normal distribution that could be used to approximate the distribution of .

Use the distribution from (c)(ii) to approximate the probability that in at least one half of the 1000 spins the ball lands on a black number.

As part of a marketing promotion, 47% of packets of a particular brand of crisps contain a zombie toy as a prize.  A random sample of 100 packets is taken.

Find the exact value of the probability that exactly 49 of the packets contain a prize. Give your answer to 6 decimal places. 

Write down the normal distribution that could be used to approximate the distribution for the number of the 100 packets that contain a prize.

The weight,  of a chocolate bar produced by a certain manufacturer is modelled as

Find 

Hilda takes a random sample of chocolate bars and weighs them. Find the probability that the 8^th chocolate bar that Hilda takes is the first one that weighs less than .

Zelda buys a pack containing 12 of the chocolate bars. It may be assumed that the 12 bars in the pack represent a random sample. Find the probability that exactly two chocolate bars in the pack weigh less than .

State the conditions that must be satisfied to be able to model a random variable  with a binomial distribution

A fair spinner has 5 sectors labelled with the numbers 1 through 5. The spinner is spun and a fair coin is flipped, and the number the spinner lands on along with the result of the coin flip (heads or tails) are recorded.  For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

If the coin lands on heads, then the random variable  is the number of the sector that the spinner lands on times two.  Otherwise is the number of the sector that the spinner lands on plus 10.

The random variable is the number of times the spinner is spun and the coin is flipped until an odd number on the spinner occurs together with tails on the coin.

The random variable  is the number of times a prime number on the spinner occurs together with heads on the coin, when the spinner is spun and the coin is flipped 21 times.

Each time the spinner is spun and the coin is flipped, it is a ‘win’ if a square number on the spinner occurs together with heads on the coin, or it is a ‘loss’ if a non-square number on the spinner occurs together with tails on the coin. Any other outcome is a ‘draw’.  The random variable  is the number of losses when the spinner is spun and the coin is flipped twelve times.

For the random variable  defined in (b)(i) above, give the name of the probability distribution that would be appropriate for modelling . Give the values(s) of any parameters.

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

A cake recipe calls for a certain amount of flour to be used. The random variable is the number of cakes that can be made, following the recipe exactly each time, from a bag containing a random amount of flour.

A student cuts a one-metre length of rope into two pieces at a random point. The random variable is the difference in length between the two pieces of rope that result.

People are chosen at random from the UK population. The random variable  is the age of a randomly selected person, measured from their date and time of birth.

Each of the following histograms shows the distribution of results when a large number of measurements of the random variables ,  or are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable.  Where a normal model is appropriate, suggest a real-world variable that might show such a distribution.

State the conditions that must be met for the distribution of a binomial random variable to be able to be approximated by a normal random variable.

For each of the following binomial random variables, :

• if  can be approximated by a normal distribution, then write down the normal approximation to  in the form  , giving the values of and ,
  
  
• if  cannot be approximated by a normal distribution, then give a reason why.

The random variable  .

Give two reasons why a normal distribution can be used to approximate .

Find, using the appropriate normal approximation:

Using the normal approximation, find the largest value of  (where  is an integer) such that 

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution

On a European-style casino roulette wheel, the probability of the ball landing on a red number is .

The wheel is spun 36 times, and the ball lands on a red number  times.

Find 

In a separate experiment, the wheel is spun 1000 times and , the number of times the ball lands on a red number, is recorded.

Write down the normal distribution that could be used to approximate the distribution of .

Use the distribution from (c)(ii) to approximate the probability that in 1000 spins the ball lands on a red number either less than 482 times or more than 491 times.

Due to a manufacturing irregularity, 41% of Adventure Dude action figures were produced with two left hands.  Although not especially rare, and therefore not especially collectible, these so-called ‘double left’ figures are nonetheless considered to be collector’s items by hard-core Adventure Dude fanatics.

A vintage toy shop has obtained 100 Adventure Dude action figures.  These may be assumed to represent a random sample.

Find the exact value of the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer to 6 decimal places.

Use an appropriate normal approximation to approximate the probability that exactly 45 of the 100 figures are ‘double left’ figures.

Find the percentage error when using your normal approximation from part (b) to estimate the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer correct to two decimal places.

The weight,  kg, of the feed in a sack of partridge feed produced by a certain manufacturer is modelled as  .

Find 

Roger buys ten sacks of the manufacturer’s partridge feed to feed to the partridges who have begun showing up at his backyard bird feeding station. 

Find the probability that all ten sacks contain feed with a weight that is within .

Find the probability that at least two of the ten sacks are not within .

State the conditions that must be satisfied to be able to model a random variable  with a binomial distribution .

A fair spinner has 5 sectors labelled with the numbers 2, 3, 5, 7 and 11. A fair dice has 6 sides labelled with the numbers 1 through 6.  The spinner is spun and the dice is rolled, and the numbers that the spinner and the dice land on are recorded.  For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

The random variable  is the square root of the number that the spinner lands on, times the number that the dice lands on.

The random variable  is the number of times that both the spinner and the dice land on a prime number, when the spinner is spun and the dice is rolled 47 times. 

The random variable  is the number of times the spinner is spun and the dice is rolled until the number the spinner lands on is a factor of the number the dice lands on. 

On the first spin of the spinner and roll of the dice, it is a ‘win’ if the number on the spinner is greater than the number on the dice. On subsequent spins of the spinner and rolls of the dice, it is only a ‘win’ if the number the spinner lands on is higher than both the number the dice lands on and all the numbers that the dice has landed on previously.  The random variable   is the number of ‘wins’ when the spinner is spun and the dice is rolled 24 times.

For the random variable  defined in (b)(i) above, give the name of the probability distribution that would be appropriate for modelling . Give the values(s) of any parameters.

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

A string collector decides to measure all the pieces of string in his collection. The random variable is the length of a randomly chosen piece of string from the collection, rounded to the nearest centimetre.

A random sample of 1000 people is chosen from the US population, and the number of siblings each has is recorded. The random variable  is the mean number of siblings for the 1000 people in the sample. 

The masses of individual eggs in the nests of a particular species of bird are measured and recorded. The random variable  is the total mass of the eggs in a randomly chosen nest.

Each of the following histograms shows the distribution of results when a large number of measurements of the specified random variables –, or  – are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable, and suggest a real-world variable that might show such a distribution.

State the conditions that must be met for the distribution of a binomial random variable to be able to be approximated by a normal random variable.

For each of the following binomial random variables, :

• if  can be approximated by a normal distribution, then write down the normal approximation to  in the form  , giving the values of and .
• if  cannot be approximated by a normal distribution, then give a reason why

The random variable  .

 Explain why a normal distribution can be used to approximate .

Find, using the appropriate normal approximation:

Using the appropriate normal approximation, find the smallest value of (where )  such that  .

On the roulette wheel at the Dunes Oddstacker casino, the probability of the ball landing on a red number is , and the probability of the ball landing on a black number is the same.  In general, the majority of bettors will lose their bets if the ball lands neither on a red number nor on a black number.

The wheel is spun 50 times, and the ball lands on a number that is either red or black  times.

Find 

In a separate experiment, the wheel is spun 1000 times and , the number of times the ball lands neither on a red number nor on a black number, is recorded.

Explain why a normal approximation would be appropriate in this case.

Use an appropriate normal distribution to approximate the probability that in 1000 spins the number of times the ball lands neither on a red number nor on a black number is neither 80 or more nor less than 75.

Due to a production error, 58% of Bobbie Sue dolls were manufactured with proportions that might be seen on a real human being.  Plastic surgeons have been buying up these so-called ‘realie’ dolls, out of a concern that if too many of them are seen by the general public then numbers of people seeking plastic surgery will decrease.

A toy shop has received an order of 100 Bobbie Sue dolls.  These may be assumed to represent a random sample.

Calculate the percentage error when an appropriate normal approximation is used to calculate probability that exactly 58 of thr 100 dolls are 'realie' dolls.

A machine is used to produce the barge poles sold by the You Would Touch It With One Of Ours barge pole company.  The actual length,  m, of the barge poles is normally distributed with mean  m, standard deviation  m, and interquartile range 0.01712 m. 

Twenty of the barge poles are chosen at random. 

Find the probability that at most two of the barge poles will be shorter than the mean by 1 cm or more.