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AP Statistics: Section 8.2 Geometric Probability

AP Statistics: Section 8.2 Geometric Probability.

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AP Statistics: Section 8.2 Geometric Probability

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  1. AP Statistics: Section 8.2Geometric Probability

  2. Spence has trouble getting girls to say “Yes” when he asks them for a date. In fact, only 10% of the girls he asks actually agree to go out with him. Suppose that p = 0.10 is the probability that any randomly selected girl, assume independence, will agree to out with him. Spence desperately wants a date for the prom. (a) What is the probability that at least one of the first 5 girls asked will say “yes”. (b) How many girls can he expect to ask before the first one says “yes”?

  3. In (b) we will let X = the number of times Spence needs to ask a girl for a date before a girl accepts. Why is this not a binomial distribution?

  4. A random variable that counts the number of trials needed to obtain one success is called geometric and the distribution produced by this random variable is called a geometric distribution.

  5. The Geometric Setting1. Each observation falls into one of just two categories: _________ or _________.2. The n observations are all _______________.3. The probability of success, call it __, is __________ for each observation.4. *The variable of interest is the number of trials required to obtain __________________.

  6. Example 1: Consider rolling a single die.X = the number of rolls before a 3 occurs.Is this a geometric setting?

  7. P(X = 1) = P(3 on 1st roll) = P(X = 2) = P(not 3 on 1st roll and 3 on 2nd roll) = P(X = 3) = P(not 3 on 1st or 2nd roll and 3 on 3rd roll) = P(X = 4) = P(not 3 on 1st, 2nd and 3rd roll and 3 on 4th roll) =

  8. Rule for Calculating Geometric ProbabilitiesIf X has a geometric distribution with probability p of success and (1 – p) of failure on each observation, the possible values of X are 1, 2, 3, . . . . 8If n is any one of these values, the probability that the first success occurs on the nth trial is:

  9. TI83/84:

  10. Example: What is the probability that the 6th girl Spence asks to the prom will say “yes?”

  11. Construct a probability distribution table for X= number of rolls of a die until a 3 occurs.Note that the number of table entries for X will be infinite. The probabilities are the terms of a geometric sequence, _______________, hence the name for this random variable.

  12. As with all probability distributions, the sum of the probabilities must be ___.Recall from Algebra II, maybe Pre-Calculus, thatthe sum of a geometric sequence is _________. So…

  13. In the probability histogram, the first bar represents the probability of ________. The height of all subsequent bars is smaller since you are multiplying by a number less than 1. So the histogram will be _____-skewed. Always.

  14. The Mean and Standard Deviation of the Geometric Random VariableIf X is a geometric random variable with probability of success p on each trial, then

  15. Example 2: A game of chance at the state fair involves tossing a coin into a saucer. You win a stuffed animal if the coin lands in and stays on the saucer. A person wins on average 1 out of every 12 times she/he plays. What is the expected number of tosses for a win? What is the standard deviation?

  16. P(X > n)The probability that it takes more than n trials to see the first success is ________Example 3: What is the probability that it takes more than 12 tosses to win a stuffed animal?

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