Probability Distributions (DP IB Maths: AI HL)

Frank plays a game involving a biased six-sided die.

The faces of the die are numbered 1 to 6.

The score of the game,X , is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

Score, x	1	2	3	4	5	6
						3p

Calculate the exact value of p.

Frank plays the game once.

Calculate the expected score.

Frank plays the game twice and adds the scores together.

Find the probability Frank has a total score of 4, giving your answer as a fraction.

A weekly raffle ticket costs , with three different levels of prizes, . The grand prize in the first week is $100 and it increases by $5 every week if nobody wins it.

The following table shows the probability distribution for S.

Prize, s	0	20	Grand prize
	9p	7p	4p

Find the value of p.

Given the grand prize is not won, write down an expression for the grand prize, G, in the form , where a and b are constants to be found and n is the week of the raffle.

Given the raffle is a fair game in the fourth week, find the value of k.

Find the week in which the expected profit for the ticket buyer is $5.

A shooting target is divided in to three regions A, B and C. Contestants pay $7.50 to enter and get to take one shot.

The probability of hitting each region is given in the following table:

Region	A	B	C	Missed target
Probability

It is given  and .

Find the value of a and b.

A contestant’s prize is dependent on the target they hit.

Calculate the value of such that the game is a fair game.

A game is played where contestants shoot a football at a goal with a goal keeper.  The goal is divided into five regions; A, B, C, D and E.  Each region is assigned with the scores, X, outlined in the table below.

The following table shows the value of X for each region and the probability distribution for X.

Region	A	B	C	D	E	Miss
Score	1	4	4	8	8	-2
	0.3	p	p	q	q	0.4

Find the exact value of  p and  q.

Calculate the expected score.

Find the probability that a player has a score of 16 after two rounds.

A spinner has six sections; A, B, C, D, E and F.  The table below shows the area of the spinner occupied by each section and their respective pay offs.

Section	A	B	C	D	E	F
Area
Prize	$6	$5	$4	$3	$2	$1

Calculate the value of p.

The game costs $4 and John says that the expected profit from playing the game is $0.30.

Calculate the percentage error in John’s claim.

A biased coin has a probability of showing tails as 0.85. Leon plays a game where he flips the coin.  He pays $15  to play.  If the coin lands on tails he receives nothing but if it lands on heads he receives $5c.The game is fair.

Determine the value of c and write down the total prize if he wins.

Two biased coins are tossed and a fair spinner divided into four equal sectors numbered 1 to 4 is spun.

Write down the total number of possible outcomes when the two coins are tossed and the spinner is spun.

A random variable, , is defined as the number of tails when the two coins are tossed multiplied by the number the spinner lands on when it is spun.

For each coin the probability of getting tails is  .

Complete the following probability distribution table for X:

x	0	1	2	3	4	6	8

Represent the probability distribution for as a piecewise function in the form:

A discrete random variable X has the following probability distribution:

x	-5	-3	-1	0	1	3	5
	0.24	2k2	0.04	0.12	3k	0.19	0.11

Find the value of k.

Find .

A discrete random variable X has the probability distribution shown in the following table: 

x	0	1	2	3	4	5	6
	a	2a	3a	4a	5a	6a	7a

 Find the value of a.

Find:

(i)

(ii)

(iii)

(iv)

Tom has constructed a biased spinner with six sectors labelled 1 to 6.

The probability of the spinner landing on each of the six sectors is shown in the following table:

Number on sector	1	2	3	4	5	6
probability	1.5p	p

Find the value of p.

Tom is playing a game with his biased spinner. The score for the game, X, is the number which the spinner lands on after being spun.

Tom plays the game twice and adds the two scores together. Find the probability that Tom has a total score of 9.

Find the probability that X is 

Ben plays a game involving a biased eight-sided die.

The faces of the die are labelled -4, -2, -1, 0, 1, 3, 5, 6.

The score of the game, X, is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

Score, x	-4	-2	-1	0	1	3	5	6
		p

It is given that .

Calculate the exact values of p and q.

Ben plays the game once.

Calculate the expected score.

Ben plays the game twice and adds the scores together.

Find the probability that Ben has a total score of -3, giving your answer as a fraction in its simplest form.

A weekly lottery ticket costs $15, with five different levels of prizes, $s. The grand prize in the first week is $2000 and it increases by 20% each week that nobody wins it.

The following table shows the probability distribution for S.

Prize, $s	0	2	10	20	100	Grand prize
					p

Find the value of p.

Determine if the lottery is a fair game in the first week.

Given the grand prize is not won, write an expression in terms of for the value of the grand prize in the nth week of the lottery.

The wth week is the first week which a player is expected to make a profit.

Calculate the value of w and the expected profit. Give your answer correct to 2 decimal places.

A discrete random variable X  has the following probability distribution:

x	-5	-3	-1	0	1	3	5
	0.42		0.05	0.21		0.09	0.07

Find the value of k.

Find .

A discrete random variable X has the following probability distribution:

x	0	1	2	3	4
	0.04	0.35	a	0.21	b

The value of .

Write down two equations connecting a and b.

Hence find the value of a and the value of b.

Find 

Two fair six-sided dice are rolled. One is a standard die with sides numbered 1 to 6 and the other die has sides numbered 1, 1, 2, 2, 4, 4.

The discrete random variable S is the sum of these two dice when they are rolled.

Complete the following probability distribution table.

s

Find .

Let X be the discrete random variable represented in the probability distribution table below.

x	0	1	2	3	4	5	6	7	8
	0.32	0.22	0.21

Find the value of k.

Find the expected value of .

The table below represents the number of pets and the corresponding probability of a house having that number of pets.

Number of pets, x	0	1	2	3	4
	0.44	0.21	0.19	p	0.02

Find the value of p.

Find the expected number of pets in a house.

There was a recording error and houses with 5 pets were not counted. It is found that there are 6 houses with 5 pets. The neighbourhood in total has 406 houses, including these 6 houses.

Complete the following table for the true probability distribution of the number of pets, x.

Number of pets,  x	0	1	2	3	4	5

Find the actual expected number of pets.

Calculate the percentage error between your answer in part (b) and your answer in part (d).

The table below represents the probability distribution for the number of apple products people have in a city in France.

Number of apple products, x
0	0.1925
1	0.1815
2	0.2250
3	p
4	0.0895
5	0.0504
6	0.0307
7	0.0104
8	q

It is given that    

Find the value of  p and q.

Find the expected number of apple products a randomly selected person from this city has.

The city has a population of 412 000.  The average amount someone from this city spends on an apple product is €825.

Estimate the revenue apple earned from sales to people in this city. Give your answer to the nearest euro (€).

A discrete random variable X has the probability distribution shown in the following table:

x	-5	-1	2	6
			p	4p

Find the value of p.

X is sampled twice such that the results of the two experiments are independent of each other and the outcomes of the two experiments are recorded.  A new random variable, Y, is defined as the sum of the two outcomes .

Draw a probability distribution table for Y.

The table below represents the number of strokes Josh takes on a particular round of golf and the corresponding frequencies over a year of playing at the same golf course.

Complete the following probability distribution table for the data above.3

Par refers to the number of strokes a golfer is expected to need to complete the play on a golf course. The par number of strokes for Josh’s golf course is 72.

Determine whether Josh’s expected number of strokes is less than or greater than the par number of strokes.