Samantha is a baseball player, she has a batting average of 0.280.
a) find the probability that at least 3 hits in her next 5 times at bat (12 marks)
b)what is Samantha's expected number of hits in her next 10 times at bat? (3 marks)
Use binomial distribution, p=0.28, q=(1-0.28)=0.72
A) n=5
P(3/5)+P(4/5)+P(5/5)
=3C5*p^3*q^2 + 4C5*p^4*q^1 + 5C5*p^5*q^0
=10(0.28^3)(0.72^2) + 5(0.28^4)(0.72) + 1(0.28^5)(0.72^0)
Use your calculator to find the answer for (A)
B)
Expected value for binomial distribution with n=10, p=0.28, q=0.72 is npq
=10*0.28*0.72
a) To find the probability that Samantha gets at least 3 hits in her next 5 times at bat, we can use the binomial distribution formula.
The binomial distribution formula is:
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 out of n trials,
p is the probability of success in a single trial, and
(1-p) is the probability of failure in a single trial.
n represents the number of trials.
In this case, Samantha's success is getting a hit, so her probability of success is her batting average, which is 0.280. The number of trials is 5 (her next 5 times at bat). We need to find the probability of getting at least 3 hits, so we need to calculate the probabilities of getting 3, 4, and 5 hits and sum them up.
Let's calculate the probabilities:
P(X=3) = C(5, 3) * 0.280^3 * (1-0.280)^(5-3)
P(X=4) = C(5, 4) * 0.280^4 * (1-0.280)^(5-4)
P(X=5) = C(5, 5) * 0.280^5 * (1-0.280)^(5-5)
To calculate the probabilities, we need to know the value of C(n, k). This can be calculated using the formula:
C(n, k) = n! / (k! * (n-k)!)
Using this formula, we can find the probabilities for each value of k and sum them up to get the probability of getting at least 3 hits.
b) To find Samantha's expected number of hits in her next 10 times at bat, we can multiply her batting average (0.280) by the number of times at bat (10).
Expected number of hits = batting average * number of times at bat
Expected number of hits = 0.280 * 10 = 2.8 hits
a) To find the probability that Samantha gets at least 3 hits in her next 5 times at bat, we can use the binomial probability formula. The formula is:
P(X >= k) = 1 - P(X < k)
where P(X < k) is the cumulative probability of getting less than k hits in the given number of trials.
Using this formula, we can calculate the probability as follows:
Number of trials (n) = 5
Probability of success (p) = batting average = 0.280
Number of successes (k) = 3, 4, or 5 (since we're calculating the probability of getting at least 3 hits)
Using the binomial probability formula, we can find the cumulative probability:
P(X >= 3) = 1 - P(X < 3)
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
For Samantha's case:
P(X = 0) = C(5, 0) * 0.280^0 * (1 - 0.280)^(5 - 0)
= 1 * 1 * 0.72^5
= 0.16807
P(X = 1) = C(5, 1) * 0.280^1 * (1 - 0.280)^(5 - 1)
= 0.280 * 0.72^4
= 0.30294
P(X = 2) = C(5, 2) * 0.280^2 * (1 - 0.280)^(5 - 2)
= 0.280^2 * 0.72^3
= 0.30294
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
= 0.16807 + 0.30294 + 0.30294
= 0.77395
P(X >= 3) = 1 - P(X < 3)
= 1 - 0.77395
= 0.22605
Therefore, the probability that Samantha gets at least 3 hits in her next 5 times at bat is 0.22605.
b) To find Samantha's expected number of hits in her next 10 times at bat, we can multiply the number of trials (n) by the probability of success (p).
Number of trials (n) = 10
Probability of success (p) = batting average = 0.280
Expected number of hits = n * p
= 10 * 0.280
= 2.8
Therefore, Samantha's expected number of hits in her next 10 times at bat is 2.8.