Chapter 8 Day 1

The Binomial Setting - Rules 1. Each observations falls under 2 categories we call success/failure (coin, having a child, cards – heart vs. not a heart) 2. There is a fixed number of n observations. 3. The n observations are independent (knowing the result of 1 observation has no effect on future observations. 4. The probability, p, is the same for each observation.

Examples – Is it reasonable to use a binomial distribution for the random variable X? Deal 10 cards from a shuffled deck and count the number X of red cards. ▫No, if the first card flipped is a black card, there is a greater chance of a red on the second flip because there are more reds in the deck. So, observations are NOT independent.

Examples – Is it reasonable to use a binomial distribution for the random variable X? Blood type is inherited. If both parents carry genes for the O and A blood types, each child has a probability of 0.25 of getting two O genes and so of having blood type O. The number of O blood types among 5 children of these parents is the count X. ▫Yes! There are two possible outcomes: type O and not type O. There is a fixed number of observations – 5 children. Different children inherit independently of one another. The probability is 0.25 for each child.

Binomial Distribution The distribution of the count X of successes in the binomial setting is the binomial distribution with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers 0 to n. As an abbreviation, we say that X is B(n,p).

Example Getting a head from 10 flips of a coin. ▫X = the outcomes of getting a head ▫n = ▫p = ▫Notation:

Binomial Coefficient The number of ways of arranging k successes among n observations is given by the binomial coefficient: for k = 0, 1, 2, …, n

Binomial Probability If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2,…, n. If k is any one of these values,

Example Jack burns 15% of all pizzas cooked. He cooks 9 pizzas. ▫Write the correct binomial distribution notation. ▫What is the probability that he burns 3 pizzas? ▫What is the probability that he burns at most 3 pizzas?

P.D.F. Given a discrete random variable X, the probability distribution function assigns a probability to each value of X. Calculator function – binompdf(n,p,x) calculates the binomial probability without the long formula! ▫To find it hit 2 nd, DISTR

Example Back to the pizza problem… Use the calculator commands to solve the problem! Find the probability of burning 3 pizzas. Find the probability of burning at most 3 pizzas.

C.D.F. Given a random variable X, the cumulative distribution function of X calculates the sum of the probabilities for 0, 1, 2,…, up to the value X. That is, it calculates the probability of obtaining at most X successes in n trials. On the calculator, binomcdf(n, p, x) ▫To find it hit 2 nd, DISTR Try it on the pizza problem – what is the probability of burning at most 3 pizzas?

Example – HW Question #3 Each child born to a particular set of parents has probability of 0.25 of having blood type O. Suppose these parents have 5 children. Let X = number of children who have type O blood. Then X is B(5,0.25) A. What is the probability that exactly 2 children have type O blood? (show the calculator set up) B. Make a table for the PDF of the random variable X. Then use the calculator to find all possible values of X and complete the table. ▫Use a list to help!

C. Verify that the sum of the probabilities is 1. D. Construct a histogram of the PDF. Use the calculator to find the cumulative probabilities, and add these values to your PDF table. Then construct a cumulative histogram. ▫Use a list to help!