Mike Vanderjagt is the most accurate kicker in the history of professional football. He makes 91.2% of his field goal attempts. What is the probability that he will miss at most 1 out of the next 4?
prob(miss none) = C(4,0)(.088)^0 (.912)^4 = ...
prob(miss one) = C(4,1)(.088)(.912)^3 = ..
add them up
p success = .91
p fail = .09
chances of missing zero + chances of missing one
4C0 = 1
so
1 * .09^4 * .91^0 = .09^4 = 6.56*10^-5
4C1 = 4!/[3!] = 4
so
4*.09^3*.91 = .00265
.00265 +.0000656 = .00272
To find the probability that Mike Vanderjagt will miss at most 1 out of the next 4 field goal attempts, we can use the binomial probability formula.
The binomial probability formula is:
P(x) = (nCx) * p^x * q^(n-x)
Where:
P(x) is the probability of getting exactly x successes,
n is the total number of trials,
x is the number of desired successes,
p is the probability of success in a single trial, and
q is the probability of failure in a single trial.
In this case:
n = 4 (the total number of field goal attempts)
x = 0 or 1 (the number of desired successes - Mike missing 0 or 1 field goal attempts)
p = 0.912 (probability of success - Mike making a field goal)
q = 1 - p = 1 - 0.912 = 0.088 (probability of failure - Mike missing a field goal)
Now, let's calculate the probability.
For x = 0 (Mike misses 0 field goals):
P(0) = (4C0) * 0.912^0 * 0.088^(4-0)
P(0) = (1) * 1 * 0.088^4
P(0) = 1 * 0.088^4
P(0) = 0.088^4
For x = 1 (Mike misses 1 field goal):
P(1) = (4C1) * 0.912^1 * 0.088^(4-1)
P(1) = (4) * 0.912^1 * 0.088^3
P(1) = 4 * 0.912 * 0.088^3
Finally, to find the probability that Mike will miss at most 1 out of the next 4 field goal attempts, we add the probabilities of Mike missing 0 and 1 field goals together:
P(at most 1) = P(0) + P(1)
P(at most 1) = 0.088^4 + 4 * 0.912 * 0.088^3
Now, you can calculate this expression to find the answer.