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Essential Questions. How do we use the Binomial Theorem to expand a binomial raised to a power? How do we find binomial probabilities and test hypotheses?.
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Essential Questions • How do we use the Binomial Theorem to expand a binomial raised to a power? • How do we find binomial probabilities and test hypotheses?
A binomial experimentconsists 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:
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 3C2 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.
The probability that Jean will make each free throw is , or 0.5. 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? P(r) = nCr prqn-r Substitute 3 for n, 1 for r, 0.5 for p, and 0.5 for q. P(1) = 3C1 (0.5)1 (0.5)3-1 The probability that Jean will make exactly one free throw is 37.5%.
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) + 3C2(0.5)2(0.5)3-2 + 3C3(0.5)3(0.5)3-3 3C1(0.5)1(0.5)3-1 The probability that Jean will make at least one free throw is 87.5%.
The probability that the counselor will be assigned 1 of the 3 students is . Substitute 3 for n, 2 for r, for p, and for q. Example 3: Finding Binomial Probabilities 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? The probability that Counselor Jenkins will get 2 of the next 3 students assigned is about 22%.
Example 4: Finding Binomial Probabilities 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? At least 2 answers correct is the same as exactly 2, 3, 4, or 5 questions correct. The probability of answering a question correctly is 0.25. P(2) + P(3) + P(4) + P(5) 5C2(0.25)2(0.75)5-2 + 5C4(0.25)4(0.75)5-4 + 5C3(0.25)3(0.75)5-3 + 5C5(0.25)5(0.75)5-5
