Answer:In the Monty Hall game show, there are three doors that hide either a goat or a car. Two of the doors hide a goat and the remaining door hides the car. In the game, each player is asked to choose one door at random. Suppose that there are 15 players playing the game in a particular month.
a) Find the expected number of players who win a car
b) Find the probability that
i) exactly five players win a car
ii) more than five players win a car
Step-by-step explanation: a) Find the expected number of players who win a car:
Each player has a 1/3 chance of winning a car and a 2/3 chance of winning a goat because there are three doors. Since the players' choices are independent, you can treat each player's choice as a Bernoulli trial with a success probability of 1/3 (winning a car).
The expected number of players who win a car follows a binomial distribution since we have multiple independent trials (players) with the same success probability.
Let X be the number of players who win a car. Then, X ~ Binomial(15, 1/3) because there are 15 players.
To find the expected value (mean) of a binomial distribution, you can use the formula:
E(X) = n * p
Where:
n is the number of trials (in this case, 15 players).
p is the probability of success in each trial (1/3).
E(X) = 15 * (1/3) = 5
So, the expected number of players who win a car is 5.
b) Find the probability that:
i) Exactly five players win a car:
To find the probability that exactly five players win a car, you can use the probability mass function (PMF) of the binomial distribution:
P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))
Where:
n is the number of trials (15 players).
k is the specific number of successes (in this case, 5 players).
p is the probability of success in each trial (1/3).
P(X = 5) = (15 choose 5) * ((1/3)^5) * ((2/3)^(15-5))
You can calculate (15 choose 5) using the binomial coefficient formula:
(15 choose 5) = 15! / (5! * (15 - 5)!) = 3003
Now, calculate P(X = 5):
P(X = 5) = 3003 * ((1/3)^5) * ((2/3)^(10))
ii) More than five players win a car:
To find the probability that more than five players win a car, you need to calculate the complementary probability of fewer than or equal to five players winning a car:
P(X > 5) = 1 - P(X ≤ 5)
You've already calculated P(X = 5) in part b(i). Now, calculate P(X ≤ 5) by summing the probabilities for k = 0, 1, 2, 3, 4, and 5:
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Then, subtract this value from 1 to find P(X > 5):
P(X > 5) = 1 - P(X ≤ 5)
You can calculate these probabilities using the binomial PMF formula as shown earlier for P(X = 5).