Unit 6 Random Variables

A model rocket kit consists of two parts: engine and rocket. Example (Review) A model rocket kit consists of two parts: engine and rocket. The mass of the rocket is approx N(134, 4) grams. The mass of the engine is approx N(36, 2) grams. Find the prob. that a kit weighs more than 195 grams.

6.3A Binomial Random Variables Determine if binomial conditions have been met Compute and interpret binomial probabilities

Conditions for a Binomial Setting: Binary—the possible outcomes are only success and failure Independent—Trials must be independent (knowing that one trial has occurred….) Number—the number of trials n is fixed in advance Success—on each trial the prob p of success must be the same

Examples: Roll a fair die 10 times and let X = the number of sixes. Shoot a basketball 20 times from various distances. Let Y = # shots you make. Observe the next 100 cars that go by and let C = color

Example: Rolling Sixes In a game involving dice, it is desirable to roll a six. P(six) = 1/6. If X = # of sixes in four rolls, then X is a binomial with n = 4 and p = 1/6

Example: Rolling Sixes B(n, p) =B(4, 1/6) (Note: You must check conditions explicitly, define the variable X, and list the parameters involved.)

Example: Rolling Sixes What is P(X= 0)?

Example: Rolling Sixes What is P(X= 1)?

Example: Rolling Sixes What is P(X= 2)?

Example: Rolling Sixes What is P(X= 3)?

Example: Rolling Sixes What is P(X= 4)?

General Formula for Binomial Probabilities 𝑃 𝑿=𝑘 = 𝑛 𝑘 𝑝 𝑘 (1−𝑝) 𝑛−𝑘 Note: Some people substitute 𝑞 for (1−𝑝)

Example: Roulette In Roulette 18 of the 38 spaces on the wheel are black. Suppose you observe the next 10 spins of the wheel. What is the probability that exactly half of the spins land on black? What is the probability that at least 8 of the spins land on black?

Using TI Technology: binompdf(n, p, k) computes 𝑃(𝑋=𝑘) binomcdf(n, 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 binomial conditions.

If we sample without replacement do we have independence? Special Note: Independence If we sample without replacement do we have independence?

Consider: Drawing three cards from a standard deck. Success = Black Card Let E = first card black Let F = second card black Let G = third card black

Consider: Drawing three cards from one standard deck. Success = Black Card What is the probability of drawing a black card, given that the first two cards were black… With replacement Without replacement

Consider: Drawing three cards from 100 standard decks. Success = Black Card What is the probability of drawing a black card, given that the first two cards were black… With replacement Without replacement

Absolute independence The probabilities found with replacement are exactly the same as those found without replacement (In other words, we meet the def of indep found in the last unit.)

Practical independence Sampling without replacement results in probabilities not significantly different from those found sampling with replacement.