Section 6.3 – Binomial Distributions
Some experimental situations have two outcomes:
*Flipping a coin, making a free throw, the gender of a baby, etc.
Discrete
The Binomial Setting: Use BINS
1. Binary? 2 possible outcomes, success/failure
2. Independent?, Trials must be independent
3. Number? Must be a fixed number of trials
4. Success? Same probability p of success on each trial.
*Binomial Distribution  how many successes in n trials if probability of success is p. *n & p are parameters.
Notation: B(n, p) Binomial distribution with n trials and probability p of success
Ex. Plinko: 40 trials, 0.37 probability of success  B(40, 0.37)
Example, p. 388. Is it a binomial setting?
(a) Genetics says that children receive genes from each of their parents independently. Each child of a
particular set of parents has probability 0.25 of having O blood. Suppose these parents have 5 children. Let X =
the number of children with type O blood.
Is this binomial?
1. Binary? “Success” = has type O blood, “Failure” = doesn’t have type O blood
2. Independent? Each child’s blood should be independent.
3. Number? n = 5 trials (children)
4. Success? The probability of success is p = 0.25 for each trial
Yes, this situation is binomial.
(b) Shuffle a deck of cards. Turn over the first 10 cards, one at a time. Let Y = the number of aces you
observe.
Is this binomial?
1. Binary? “Success” = get an ace, “Failure” = don’t get an ace
2. Independent? No. If the first card you turn over is an ace, the next card is less likely to be an ace.
3. Number? n = 10 trials (cards)
4. Success? The probability of success is p = 4/52
No, this situation is NOT binomial because the cards are not independent.
(c) Shuffle a deck of cards. Turn over the top card. Put the card back in the deck, and shuffle again. Repeat
this process until you get an ace. Let W = the number of cards required.
Is this binomial?
1. Binary? “Success” = get an ace, “Failure” = don’t get an ace
2. Independent? Because drawing with replacement and shuffling, each trial should be independent
3. Number? Not a set number!
4. Success? The probability of success is p = 4/52 for each trial.
No, this situation is NOT binomial because there is not a set number of trials.
Probability distribution function (pdf): given a discrete random variable X, each value of X is assigned a
probability.
Example, p. 390: Inheriting Blood Type
Genetics says that children receive genes from each of their parents independently. Each child of a particular
set of parents has probability 0.25 of having O blood. Suppose these parents have 5 children. Let X = the
number of children with type O blood.
What is P(X = 0), meaning what is the probability that none of the five children have type O blood?
We already know that X is Binomial
X = # of children with type O blood, X is B(5, 0.25)
Probability that none of the children have type O blood:
P(X = 0) = P(FFFFF) = (0.75)(0.75)(0.75)(0.75)(0.75) = (0.75)5 = 0.2373
P(X = 1)? Different combinations  SFFFF, FSFFF, FFSFF, FFFSF, FFFFS
Let’s look at P(SFFFF) = (0.25)(0.75)(0.75)(0.75)(0.75) = (0.25)(0.75)4.
Since there are five possible combinations: P(X=1) = P(exactly one child has type O blood) = 5 * (0.25)(0.75)4.
= 0.39551
Refer to calculator sheet on how to do this in the calculator.
BINOMIAL PROBABILITY FORMULA: (on AP formula chart)
If X has the binomial distribution with n trials and probability p of success on each trial, the possible values
of X are 0, 1, 2, …, n. If k is any one of these values,
𝑛
𝑃(𝑋 = 𝑘) = ( ) 𝑝𝑘 (1 − 𝑝)𝑛−𝑘
𝑘
𝑛
𝑛!
Where the binomial coefficient ( ) = 𝑘!(𝑛−𝑘)!.
𝑘
HW p. 410 # 69, 70, 73, 75, 77, 102
Due: Friday