Question

A game show airs on television five days per week. Each day, a prize is randomly placed behind one of two doors. The contestant wins the prize by selecting the correct door. What is the probability that exactly two of the five contestants win a prize during a week.

Short answer

The probability that exactly two out of the five contestants will win a prize during a week is 0.3125.

Step-by-step solution

Define the parameters for the Binomial Distribution

The number of successes sought, k, is 2 (we want exactly two contestants to win). The number of trials, n, is 5 (a week has five days). The probability of success on any given trial, p, is 0.5 (there are two doors, and one prize).

Use the formula for Binomial Distribution

The formula for Binomial Distribution is given by \( P(x=k) = C(n,k) * p^k * (1-p)^{n-k} \). Applying these values to the formula gives \( P(x=2) = C(5,2) * 0.5^2 * (1-0.5)^{5-2} \).

Calculate the value of the combination

The binomial coefficient, \( C(n,k) = n! / k!(n-k)! \). Substituting these gives \( C(5,2) = 5! / 2!(5-2)! = 10 \).

Substitute the combination value back into the binomial formula and solve

Now that we have the combination value, we can substitute it back into our formula: \( P(x=2) = 10 * 0.5^2 * 0.5^{5-2} \). On solving, we get a value of \( P(x=2) = 0.3125 \).

Key concepts

Probability

In simple terms, probability measures how likely an event is to happen. In our context with the game show, it is the likelihood that a contestant picks the correct door to win the prize. Under normal circumstances, when a contestant randomly picks between the two doors, the probability is 0.5, as there are always two equally possible outcomes.
Probability can be calculated using the formula: - For a single event: Probability = Number of favorable outcomes / Total number of outcomes Here, the number of favorable outcomes is 1 (choosing the door with the prize), and the total number of possible outcomes is 2 (choosing either door). Therefore, the probability for each contestant to win on a given day is 0.5.
In our exercise, since the problem uses the concept of a binomial distribution, the interest lies in calculating the probability of having exactly two winners over five trials (one for each day), given a constant single-day winning probability of 0.5.

Combinatorics

Combinatorics involves counting, arranging, and finding patterns in sets. An essential component of combinatorics is the concept of combinations. Combinations help us understand how many ways we can select items from a larger group, where order does not matter.
For the game show problem, combinations are used to determine how many ways we can have exactly 2 winners out of 5 days of show airing.The formula for combinations is given by:- Combination formula: \( C(n, k) = \frac{n!}{k! (n-k)!} \)In our situation:- \( n = 5 \) (the total number of game show days or trials)- \( k = 2 \) (the exact desired number of contestant wins, i.e., the success days)Calculating the combination:\( C(5, 2) = \frac{5!}{2! (5-2)!} \), which equals 10.
This computation tells us there are 10 different scenarios where exactly 2 of the contestants win out of the 5 days.

Probability of Success

The probability of success in each trial is an essential element within the concept of the binomial distribution. Here, 'success' refers to a contestant picking the right door and winning the prize on a game show day. For our problem:- The probability of success, \( p \), is 0.5 for each trial. This is because there's a one in two chance of picking the correct door.To determine the total probability of exactly two successes (or wins) over the week, we combine the calculated combinations with the probability of success raised to the power of successful trials, and the probability of failure raised to the power of unsuccessful trials. The formula for finding the probability of exactly \( k \) successes in \( n \) trials:- \( P(x=k) = C(n, k) * p^k * (1-p)^{n-k} \)For our specific example:- \( C(5, 2) \) is 10 (derived from combinations)- \( p = 0.5 \)Substituting these values gives the probability of exactly 2 wins out of 5 days as 0.3125. This number represents the chance that, over the course of the week, exactly two contestants will successfully pick the correct door and win the prize.