2. Holt McDougal Algebra 2 Binomial Distributions How do we use the Binomial Theorem to expand a binomial raised to a power? How do we find binomial probabilities and test hypotheses?

3. Holt McDougal Algebra 2 Binomial Distributions A binomial experiment consists of n independent trials whose outcomes are either successes or failures; the probability of success p is the same for each trial, and the probability of failure q is the same for each trial. Because there are only two outcomes, p + q = 1, or q = 1  p. Below are some examples of binomial experiments:

4. Holt McDougal Algebra 2 Binomial Distributions Suppose the probability of being left-handed is 0.1 and you want to find the probability that 2 out of 3 people will be left-handed. There are 3 C 2 ways to choose the two left-handed people: LLR, LRL, and RLL. The probability of each of these occurring is 0.1(0.1)(0.9). This leads to the following formula.

5. Holt McDougal Algebra 2 Binomial Distributions Example 1: Finding Binomial Probabilities Jean usually makes half of her free throws in basketball practice. Today, she tries 3 free throws. What is the probability that Jean will make exactly 1 of her free throws? The probability that Jean will make each free throw is, or 0.5. P(r) = n C r p r q n-r P(1) = 3 C 1 Substitute 3 for n, 1 for r, 0.5 for p, and 0.5 for q. The probability that Jean will make exactly one free throw is 37.5%. (0.5) 1 (0.5) 3-1

6. Holt McDougal Algebra 2 Binomial Distributions Example 2: Finding Binomial Probabilities Jean usually makes half of her free throws in basketball practice. Today, she tries 3 free throws. What is the probability that she will make at least 1 free throw? At least 1 free throw made is the same as exactly 1, 2, or 3 free throws made. P(1) + P(2) + P(3) 3 C 1 (0.5) 1 (0.5) 3-1 The probability that Jean will make at least one free throw is 87.5%. + 3 C 2 (0.5) 2 (0.5) 3-2 + 3 C 3 (0.5) 3 (0.5) 3-3

7. Holt McDougal Algebra 2 Binomial Distributions Students are assigned randomly to 1 of 3 guidance counselors. What is the probability that Counselor Jenkins will get 2 of the next 3 students assigned? Substitute 3 for n, 2 for r, for p, and for q. The probability that Counselor Jenkins will get 2 of the next 3 students assigned is about 22%. The probability that the counselor will be assigned 1 of the 3 students is. Example 3: Finding Binomial Probabilities

8. Holt McDougal Algebra 2 Binomial Distributions Ellen takes a multiple-choice quiz that has 5 questions, with 4 answer choices for each question. What is the probability that she will get at least 2 answers correct by guessing? The probability of answering a question correctly is 0.25. 5 C 2 (0.25) 2 (0.75) 5-2 At least 2 answers correct is the same as exactly 2, 3, 4, or 5 questions correct. P(2) + P(3) + P(4) + P(5) Example 4: Finding Binomial Probabilities + 5 C 3 (0.25) 3 (0.75) 5-3 + 5 C 4 (0.25) 4 (0.75) 5-4 + 5 C 5 (0.25) 5 (0.75) 5-5

9. Holt McDougal Algebra 2 Binomial Distributions Lesson 3.3 Practice B