pete and sean decide to raise money for a charity by having a carnival in their backyard. in one of the games that they set up, the probability that a person will win is.4. if robyn plays that game nine times, what is the probability that she wins exactly four times?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
.4^4 * .6^5 = ?
To calculate the probability that Robyn wins exactly four times, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) represents the probability of getting exactly k successes,
C(n, k) is the number of ways to choose k successes from n trials (binomial coefficient),
p is the probability of success on each trial,
(1-p) is the probability of failure on each trial,
n is the total number of trials.
In this case, n is the number of times Robyn plays the game, which is 9, and p is the probability of winning, which is 0.4. We want to find the probability of Robyn winning exactly 4 times, so k is equal to 4.
Plugging the values into the formula, we have:
P(X = 4) = C(9, 4) * 0.4^4 * (1-0.4)^(9-4)
Calculating the individual terms:
C(9, 4) = 9! / (4! * (9-4)!) = 126
0.4^4 = 0.0256
(1-0.4)^(9-4) = 0.6^5 = 0.07776
Now, substitute the values into the formula:
P(X = 4) = 126 * 0.0256 * 0.07776
Calculating:
P(X = 4) ≈ 0.2497 or 24.97%
Therefore, the probability that Robyn wins exactly four times out of nine plays is approximately 0.2497 or 24.97%.