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Binomial Distribution (DP IB Applications & Interpretation (AI))

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  • FrontThe Binomial Distribution

    What is the binomial distribution?

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  • What is the binomial distribution?

    The binomial distribution is a discrete probability distribution that counts the number of successes in a fixed number of independent trials with two possible outcomes ('success' and 'failure') and a constant probability of success.

  • What are the four conditions necessary in order to use a binomial distribution?

    The four conditions necessary in order to use a binomial distribution are:

    1. A fixed finite number of trials.

    2. Outcomes of the trials are independent.

    3. Exactly two possible outcomes per trial ('success' and 'failure').

    4. A constant probability of success.

  • What notation is used to show that a random variable X has a binomial distribution?

    The notation used to show that random variable X has a binomial distribution, is X tilde straight B open parentheses n comma space p close parentheses

    Where:

    • n is the number of trials

    • p is the probability of success

    The symbol 'tilde' means 'is distributed as'.

  • What is q sometimes used to represent in a binomial distribution?

    In a binomial distribution, q is sometimes used to represent the probability of failure.

    It is equal to 1 minus p, where p is the probability of success.

  • State the equation for the expected number of successful trials for a binomial random variable X.

    The equation for the expected number of successful trials for a binomial distribution is straight E open parentheses X close parentheses equals n p

    Where:

    • X tilde straight B open parentheses n comma space p close parentheses is a binomial random variable

    • n is the number of trials

    • p is the probability of success

    straight E open parentheses X close parentheses is also the mean of the binomial distribution straight B open parentheses n comma space p close parentheses.

    This equation is in the exam formula booklet.

  • What is the equation for the variance of a binomial random variable X?

    The equation for the variance of a binomial random variable is Var open parentheses X close parentheses equals n p open parentheses 1 minus p close parentheses

    Where:

    • X tilde straight B open parentheses n comma space p close parentheses is a binomial random variable

    • n is the number of trials

    • p is the probability of success

    This equation is in the exam formula booklet.

  • True or False?

    A 'success' in a binomial model must refer to a win (or some other 'good' outcome)?

    False.

    A 'success' in a binomial model does not have to refer to a win (or some other 'good' outcome).

    A 'success' or 'successful outcome' is just a label given to one of the two possible outcomes in a binomial trial.

    E.g. when rolling a 6-sided dice you could define 'rolling a 3' as a 'success' (then rolling a 1, 2, 4, 5 or 6 becomes a 'failure').

  • True or False?

    A binomial distribution can model scenarios with an infinite number of trials.

    False.

    A binomial distribution can not model scenarios with an infinite number of trials.

    A binomial distribution can only model scenarios with a fixed, finite number of trials.

  • What assumptions must be made when using a binomial model to describe a sample drawn from a population?

    When using a binomial model to describe a sample, it must be assumed that the population is large (compared to the sample) and that the sample is random.

  • True or False?

    A binomial distribution could be used to model the number of times a coin is flipped until it first lands on 'heads'.

    False.

    A binomial distribution can not be used to model the number of times a coin is flipped until it first lands on 'heads'.

    It could take any number of flips to get a head, so the number of trials is not fixed.

  • What three pieces of information are needed to calculate straight P open parentheses X equals x close parentheses using a calculator's binomial distribution function?

    To calculate straight P open parentheses X equals x close parentheses using a calculator's binomial distribution function, you need:

    • the x value,

    • the n value (number of trials),

    • and the p value (probability of success).

  • What is cumulative probability?

    Cumulative probability is the probability that a random variable takes on a value less than or equal to a specified value.

    E.g. straight P open parentheses X less or equal than 7 close parentheses is a cumulative probability.

  • What cumulative probability on your calculator would allow you to calculate straight P open parentheses X less than x close parentheses, when X is a binomial random variable.

    To calculate straight P open parentheses X less than x close parentheses, when X is a binomial random variable, use your calculator to find the equivalent cumulative probability straight P open parentheses X less or equal than x minus 1 close parentheses.

    E.g. for a binomial random variable, straight P open parentheses X less or equal than 3 close parentheses is the same probability as straight P open parentheses X less than 4 close parentheses.

    Note: straight P open parentheses X less than x close parentheses equals straight P open parentheses X less or equal than x minus 1 close parentheses is only true for discrete random variables (like the binomial distribution). It is not true for continuous random variables (like the Normal distribution).

  • True or False?

    straight P open parentheses X greater than x close parentheses equals 1 minus straight P open parentheses X less or equal than x close parentheses works for any random variable X.

    True.

    straight P open parentheses X greater than x close parentheses equals 1 minus straight P open parentheses X less or equal than x close parentheses works for any random variable X.

  • How can straight P open parentheses a less or equal than X less or equal than b close parentheses be calculated as a difference of two cumulative probabilities, when X is a binomial random variable?

    When X is a binomial random variable, straight P open parentheses a less or equal than X less or equal than b close parentheses equals straight P open parentheses X less or equal than b close parentheses minus straight P open parentheses X italic less or equal than a italic minus italic 1 close parentheses.

    Note: this is only true for discrete random variables (like the binomial distribution). It is not true for continuous random variables (like the Normal distribution).

    Some GDCs will calculate straight P open parentheses a less or equal than X less or equal than b close parentheses whereas some will only calculate straight P open parentheses X less or equal than b close parentheses.

  • True or False?

    straight P open parentheses 5 less than X less than 9 close parentheses equals straight P open parentheses 6 less or equal than X less or equal than 8 close parentheses for a binomial distribution.

    True.

    straight P open parentheses 5 less than X less than 9 close parentheses equals straight P open parentheses 6 less or equal than X less or equal than 8 close parentheses for a binomial distribution.

  • True or False?

    straight P open parentheses X less or equal than b close parentheses equals straight P open parentheses 1 less or equal than X less or equal than b close parentheses for a binomial distribution.

    False.

    straight P open parentheses X less or equal than b close parentheses is not equal to straight P open parentheses 1 less or equal than X less or equal than b close parenthesesfor a binomial distribution.

    For a binomial distribution, straight P open parentheses X less or equal than b close parentheses equals straight P open parentheses 0 less or equal than X less or equal than b close parentheses.

  • True or False?

    For a random variable X with the binomial straight B open parentheses n comma space p close parentheses distribution, straight P open parentheses X greater or equal than a close parentheses equals straight P open parentheses a less or equal than X less or equal than n close parentheses.

    True.

    For a random variable X with the binomial straight B open parentheses n comma space p close parentheses distribution, straight P open parentheses X greater or equal than a close parentheses equals straight P open parentheses a less or equal than X less or equal than n close parentheses.

    E.g. if X tilde B open parentheses 25 comma space 0.3 close parentheses, then straight P open parentheses X greater or equal than 7 close parentheses equals straight P open parentheses 7 less or equal than X less or equal than 25 close parentheses.

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