Unit 6 Random Variables

The weight of US men aged 19-39 is approximately normally distributed with mean of 194.7 lbs, stdev of 8.3 lbs. Likewise for women: mean of 131.4 lbs with mean of 7.5 lbs. 6 men and 6 women get on an elevator with a max weight stated to be 2000 lbs. What is the probability that the elevator will be over its maximum carrying weight?

How would your answer change if you found out that this was an express elevator exclusively serving the marriage license department in a local government facility?

6.3C Geometric Random Variables Determine if geometric conditions have been met Compute & interpret geometric probabilities

Geometric Random Variables Determine if the conditions for a geometric random variable have been met Find probabilities involving geometric random variables Find the mean of a geom rand variable

Conditions for a Geometric Setting: Binary—the possible outcomes are only success and failure Independent—Trials must be independent (the result of one trial has no effect on the results of another) Trials unknown—Key word is “until” the first success occurs. So the number of trials will NOT be fixed. Success—on each trial the prob p of success must be the same

We are going to be counting the number of trials UNTIL we encounter a success. Therefore the possible values are: X = 1, 2, 3, 4, ……

Example: Roll a die and count the number of rolls until you see a four What is the probability that it takes 3 rolls of the die until I see a four? What is the probability that it takes 8 rolls until I see a four?

General Formula for Geometric Probabilities Fail, Fail, Fail, Fail, Fail,….Success!  Do we really need a formula?

Example: Monopoly One way to get out of jail is to roll doubles. Suppose that you have to stay in jail until you do so. The probability of rolling doubles is 1/6.

Example: Monopoly Explain why this is a geometric setting. Define the geometric random variable and state its distribution. Find the probability that it takes exactly three rolls to get out of jail. Find the probability that it takes more than three rolls to get out of jail.

Using TI Technology: geometpdf(p, k) computes 𝑃(𝑋=𝑘) geometcdf(p, k) computes 𝑃 𝑋 ≤𝑘 You may NOT use calculator speak on the AP exam. You must declare and define your parameters as well as confirm the geometric conditions.

Geometric Random Variable Mean of a Geometric Random Variable (probability of success = p) 𝜇 𝑋 =𝐸 𝑋 = 1 𝑝

Example: Monopoly, part 2 On average, how many rolls should it take to escape from jail? Is this formula on the formula sheet? What is the probability that it takes longer to escape from jail? What does this probability tell you about the shape of this distribution?

All Probability Distributions Discrete Continuous “John Doe” Binomial Geometric