Question

The Los Angeles Times (December 13,1992 ) reported that what \(80 \%\) of airline passengers like to do most on long flights is rest or sleep. Suppose that the actual percentage is exactly \(80 \%,\) and consider randomly selecting six passengers. Then \(x=\) the number among the selected six who prefer to rest or sleep is a binomial random variable with \(n=6\) and \(p=0.8\) a. Calculate \(p(4)\), and interpret this probability. b. Calculate \(p(6),\) the probability that all six selected passengers prefer to rest or sleep. c. Calculate \(P(x \geq 4)\).

Short answer

The probability that exactly four passengers prefer to rest or sleep is represented by p(4). The probability that all six passengers prefer to rest or sleep is given by p(6). The probability that at least four passengers prefer to rest or sleep is given by \( P(x \geq 4) \). To find the exact probabilities, calculations need to be done as per the steps given.

Step-by-step solution

Calculation of p(4)

The formula for binomial distribution is: \( P(x) = C(n,x) * p^x * (1 - p)^(n-x) \) where C(n,x) is the combination of n items taken x at a time. To find p(4): substitute n=6, x=4 and p=0.8 into the binomial formula: \( P(4) = C(6,4) * 0.8^4 * 0.2^2 \)

Calculation of p(6)

To find p(6): substitute n=6, x=6 and p=0.8 into the binomial formula: \( P(6) = C(6,6) * 0.8^6 * 0.2^0 \)

Calculation of \(P(x \geq 4)\)

The probability that x is greater than or equal to 4, \( P(x \geq 4) \) can be computed by summing up probabilities: P(4), P(5) and P(6). So, we already have P(4) and P(6) from Steps 1 and 2. We calculate \( P(5) = C(6,5) * 0.8^5 * 0.2 \) and find \( P(x \geq 4) = P(4) + P(5) + P(6) \)