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Section 8.1 - Binomial Distributions

Section 8.1 - Binomial Distributions For a situation to be considered a binomial setting , it must satisfy the following conditions: Experiment is repeated a fixed number of trials and each trial is independent of the others There are only two possible outcomes: success (S) and failure (F).

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Section 8.1 - Binomial Distributions

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  1. Section 8.1 - Binomial Distributions • For a situation to be considered a binomial setting, it must satisfy the following conditions: • Experiment is repeated a fixed number of trials and each trial is independent of the others • There are only two possible outcomes: success (S) and failure (F). • The probability of success, P(S), is the same for each trial • The random variable, x, counts the number of successful trials

  2. Symbols and Notations for Binomial Settings n = number of trials in the sample p = P(S) probability of success in a single trial x = count of the number of successes in n trials this is called a binomial random variable A binomial experiment can be symbolized as B(n,p) The probability distribution of the successes is referred to as a binomial distribution

  3. Are these binomial experiments? • If both parents carry the genes for the O and A blood types, each child has a probability of 0.25 of getting two O genes and therefore having blood type O. 5 children of these parents are chosen to observe their blood type. Success is considered having blood type O. • Deal 10 cards from a shuffled deck and count the number, x, of red cards. Success is considered as getting a red card. • An engineer chooses a SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. The engineer counts the number, x, of bad switches (success). Choosing a SRS of size n from a population, where the population is much larger than the sample, the count of X successes in the sample is approximately B(n,p)

  4. There are several ways to find the probability of exactly k successes in n trials. One way is by the Binomial Probability Formula. P(k) = nCk pk (1 – p)n – k Using the engineer looking for defective switches example which can be approximated as B(10,0.1) … What is the probability that no more than 1 switch is defective?

  5. Example: A six sided die is rolled 3 times. What is the probability of rolling exactly one 6? Number of trials (n)? Number of successes needed (x) [# of 6’s]? Probability of success on a single trial (p)? Probability of failure on a single trial (q)? 3 1 1/6 5/6 P(k) = nCk pk (1 – p)n-k P(1) = 3C1 (1/6)1 (5/6)3-1 P(1) = 0.347

  6. As n (# of trials) gets larger, the formula becomes cumbersome to use for each possible number of successes. The alternative is to use a NORMAL DISTRIBUTION to approximate the probabilities. This can only be done when n is large. To do this, we must find the mean and standard deviation of the distribution…using m = np and s = √np(1 – p)…and then calculate a z-score for the desired probability. General Rule: A Normal Distribution can be used to approximate a Binomial Distribution if np ≥ 10and n(1-p) ≥ 10

  7. Example: Sample surveys show that fewer people enjoy shopping than in the past. A survey asked a nationwide random sample of 2500 adults if they agreed or disagreed with the statement “ I like buying new clothes, but shopping is often frustrating and time-consuming.” The population that the poll wants to draw conclusions about is all US residents aged 18 and over. Suppose the in fact 60% of all adults US residents would say they “agree” with the statement. What is the probability that 1520 or more of the sample would agree?

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