Question

At an Italian restaurant, eight friends have the choice of meat or seafood as their main course, while they may choose either red or white wine. Assuming that 70% of people order meat in their main course, 60% order white wine, and the choice of main course is independent of the choice of wine, find the probability that among the eight people, at least six choose the combination of meat and red wine.

Answer 1

The task is to calculate the probability that at least six out of eight friends choose meat and red wine, using the binomial probability formula while considering the independent probabilities of choosing meat (70%) and red wine (40%).

Step-by-step explanation:

The question asks us to find the probability that at least six out of eight friends choose the combination of meat and red wine at an Italian restaurant. Given that 70% choose meat and we can infer that 40% (100% - 60%) choose red wine, and the choices are independent, we must calculate the probability of six, seven, and eight people choosing this combination individually and then sum the results to get the total probability. We use the binomial probability formula P(X = k) = C(n, k) * pk * (1 - p)(n - k) where C(n, k) is the combination of n taken k at a time, p is the individual probability of choosing meat and red wine, and X is the random variable representing the number of people choosing meat and red wine.

To find the individual probability p of choosing meat and red wine, we multiply the independent probabilities: p = 0.70 * 0.40. Then we calculate the probabilities for 6, 7, and 8 people using the binomial formula and sum those to get the total probability.