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BINOMIAL DISTRIBUTIONS

BINOMIAL DISTRIBUTIONS.

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BINOMIAL DISTRIBUTIONS

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  1. BINOMIAL DISTRIBUTIONS • A medical center has 8 ambulances. Given the ambulance’s current condition, regular maintenance, and restocking of medical supplies, the probability of an ambulance being operational is 0.96. Find the probability that at least 6 of the 8 ambulances are operational. We will solve this problem later.

  2. BINOMIAL EXPERIMENT • A probability experiment is a binomial experiment if both of the following conditions are met: • The experiment consists of n trials whose outcomes are either successes (the outcome is the event in question) or failures (the outcome is not event in question.)

  3. BINOMIAL EXPERIMENT (CON’T) • In a binomial experiment, the trials are identical and independent with a constant probability of success, known as p, and a constant probability of failure, known as 1 – p. In other words, since there is a constant probability of success, this means that the probability of success will be the same for each and every trial of the experiment.

  4. Finding a Binomial Probability • In a binomial experiment consisting of n trials, the probability, P, of r successes (where 0 ≤ r ≤ n , p is the probability of success, and 1-p is the probability of failure) is given by the following formula: • P = nCr pr (1 – p)n-r

  5. Example: Finding a Binomial Probability • According to a survey taken by USA TODAY, about 37% of adults believe that UFOs really exist. Suppose you randomly survey 6 adults. What is the probability that exactly 2 of them believe that UFOs really exists?

  6. SOLUTION • Let p = 0.37 be the probability that a randomly selected adult believes that UFOs really exist. By survey 6 adults, you are conducting n = 6 independent trials. The probability of getting exactly r = 2 successes is: • P(r=2) = 6C2 (0.37)2 (1- 0.37)6-2 = (0.37)2(0.63)4 ≈ 0.323, or 32%

  7. Constructing a Binomial Distribution Probability Number of Believers in UFOs

  8. SOLUTION TO UFO PROBLEM • The previous slide showed the binomial distribution in a histogram. • Here are the solution values for the individual r values: • P(r = 0) = 6C0(0.37)0(0.63)6 ≈ 0.063 • P(r = 1) = 6C1(0.37)1(0.63)5 ≈ 0.220 • P(r = 2) = 6C2(0.37)2(0.63)4 ≈ 0.323 • P(r = 3) = 6C3(0.37)3(0.63)3 ≈ 0.253 • P(r = 4) = 6C4(0.37)4(0.63)2 ≈ 0.112 • P(r = 5) = 6C5(0.37)5(0.63)1 ≈ 0.026 • P(r = 6) = 6C6(0.37)6(0.63)0 ≈ 0.003 • The probability of getting at most r = 2 successes is: • P(r ≤ 2) = P(2) + P(1) + P(0) ≈ .323 + .220 + .063 ≈ 0.606 • The probability that at most 2 of the people believed that UFOs really exist is about 61%

  9. Back to the • Problem: Find the probability that at least 6 of 8 ambulances are operational. Round to the nearest tenth of a percent. • At least 6 ambulances are operational when exactly 6,7,or 8 ambulances are operational. • Find P(exactly 6) + P(exactly 7)+ P(exactly 8) • Use n = 8, p = 0.96, and 1- p = 1 – 0.96 • Use the Binomial Probability formula that we used in the UFO problem. It is on slide #4.

  10. Chicken Problem • For a science project you are incubating 12 chicken eggs. The probability that a chick is female is 0.5. • Start by calculating each binomial probability using this formula: • P (r) = nCr (0.5)r (0.5)n-r = nCr (0.5)n • Then draw a histogram of the binomial distribution based on the probability that exactly r of the chicks are female. • Finally, find the most likely number of female chicks. • What do you notice about the distribution of bars on your histogram? How does it compare with the UFO histogram?

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