The probability that Pete will catch fish on a particular day when he goes fishing is 0.8. Pete is going fishing 3 days next week.
1. The probability that Pete will catch fish on exactly one day is:
a. .008
b. .096
c. .104
d. .8
Please show work
.096
The probability that Pete will catch fish on a particular day when he goes fishing is 0.8. Pete is going fishing 3 days next week.
What is the random variable in this experiment?
To find the probability that Pete will catch fish on exactly one day out of the three days he goes fishing, we can use the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k successes
- n is the total number of trials
- k is the number of successful trials
- p is the probability of success on a single trial
In this case, n = 3 (the number of days Pete goes fishing), k = 1 (the number of successful days), and p = 0.8 (the probability of catching fish on a single day).
The binomial coefficient (n choose k), denoted as (3 choose 1), can be calculated using the formula:
(3 choose 1) = n! / (k! * (n-k)!)
(3 choose 1) = 3! / (1! * (3-1)!)
(3 choose 1) = 3! / (1! * 2!)
(3 choose 1) = 3 / 2
(3 choose 1) = 3
Putting the values into the binomial probability formula:
P(X=1) = (3 choose 1) * (0.8)^1 * (1-0.8)^(3-1)
P(X=1) = 3 * (0.8) * (0.2)^2
P(X=1) = 3 * (0.8) * (0.04)
P(X=1) = 0.096
Therefore, the probability that Pete will catch fish on exactly one day out of the three days he goes fishing is 0.096.
Hence, the correct answer is b. 0.096.
To find the probability that Pete will catch fish on exactly one day out of the three days he goes fishing, we can use the binomial probability formula.
The formula for finding the probability of "k" successes in "n" trials, where the probability of success in each trial is "p", is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
In this case, Pete is going fishing for 3 days, so n = 3, and the probability of catching fish on a particular day is p = 0.8. We want to calculate P(X = 1), which represents the probability of catching fish on exactly one day.
Using the formula:
P(X = 1) = (3C1) * (0.8)^1 * (1-0.8)^(3-1)
To calculate (3C1), which represents the number of ways to choose 1 day out of 3, we use the combination formula:
(3C1) = 3! / (1! * (3-1)!) = 3
Now substituting the values:
P(X = 1) = 3 * (0.8)^1 * (1-0.8)^(3-1)
Simplifying further, we get:
P(X = 1) = 3 * 0.8 * (0.2)^2
P(X = 1) = 3 * 0.8 * 0.04
P(X = 1) = 0.096
Therefore, the probability that Pete will catch fish on exactly one day out of the three days is 0.096.
So, the correct answer is option b. 0.096.