4. Probability
Basic Concepts of Probability
3:11 minutesProblem 5.1.4
Textbook Question
Significant For 100 births, P(exactly 56 girls) and P(56 or more girls) Is 56 girls in 100 births a significantly high number of girls? Which probability is relevant to answering that question?
Verified step by step guidance1
Step 1: Identify the type of probability distribution relevant to the problem. Since the problem involves counting the number of girls in 100 births, this is a binomial distribution. The binomial distribution is defined by two parameters: the number of trials (n = 100) and the probability of success (p = 0.5, assuming equal likelihood of a boy or girl).
Step 2: Calculate P(exactly 56 girls). Use the binomial probability formula: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( \binom{n}{k} \) is the binomial coefficient. Plug in \( n = 100 \), \( k = 56 \), and \( p = 0.5 \) to compute this probability.
Step 3: Calculate P(56 or more girls). This is the cumulative probability \( P(X \geq 56) \), which can be expressed as \( P(X = 56) + P(X = 57) + \dots + P(X = 100) \). Use the binomial formula repeatedly or leverage statistical software or tables to compute this cumulative probability.
Step 4: Determine which probability is relevant to answering the question of whether 56 girls in 100 births is significantly high. To assess significance, we typically compare \( P(X \geq 56) \) to a threshold (e.g., 0.05 for a 5% significance level). If \( P(X \geq 56) \) is less than the threshold, then 56 girls is considered significantly high.
Step 5: Interpret the results. If \( P(X \geq 56) \) is very small, it suggests that 56 or more girls in 100 births is an unusual event under the assumption of equal likelihood for boys and girls. This conclusion is based on the cumulative probability rather than the probability of exactly 56 girls.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, it can be used to determine the probability of having exactly 56 girls in 100 births, where each birth can be considered a trial with a success probability of 0.5 (assuming equal likelihood of boys and girls).
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Probability Mass Function (PMF)
The probability mass function gives the probability of a discrete random variable taking on a specific value. For the binomial distribution, the PMF can be used to calculate P(exactly 56 girls) by applying the formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.
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Cumulative Distribution Function (CDF)
The cumulative distribution function provides the probability that a random variable takes on a value less than or equal to a specific value. To determine P(56 or more girls), the CDF can be used to sum the probabilities from 56 to 100, or alternatively, 1 minus the CDF value at 55, which gives the probability of having 56 or more girls in 100 births.
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