a probability that new years is on a Saturday in a randomly chosen year is 1/7. 15 years are chosen randomly.find the probability that
1. exactly four of these years have new years day on a Saturday
2. at least 2 of these years have new years day on a Saturday
prob(Saturday) = 1/7
prob(not Saturday) = 6/7
a) prob(as stated) = C(15,4) (1/7)^4 (6/7)^11
= appr .1043
b) prob(at least 2 Saturdays)
---> 1 - (no Saturday + 1 Saturday)
= 1 - ( C(15,0) (1/6)^0 (6/7)^15 + .....)
I will let you finish that one
To find the probability in each of these scenarios, we need to use the concept of binomial probability. The binomial probability formula is:
P(x) = (nCx) * p^x * q^(n − x)
Where:
P(x) is the probability of x successes,
n is the total number of trials,
x is the number of successful outcomes,
p is the probability of a successful outcome on a single trial,
q is the probability of a failure on a single trial, which is 1 − p, and
nCx is the binomial coefficient, which represents the number of ways to choose x successes from n trials.
Let's break down each scenario:
1. Exactly four of these years have New Year's Day on a Saturday:
In this case, n = 15 (the number of years chosen randomly), x = 4 (exactly four years with New Year's Day on a Saturday), p = 1/7 (probability of New Year's Day being on a Saturday), and q = 1 − p = 6/7.
Using the binomial probability formula, we can calculate P(4):
P(4) = (15C4) * (1/7)^4 * (6/7)^(15 − 4)
2. At least 2 of these years have New Year's Day on a Saturday:
To find the probability of at least 2 years with New Year's Day on a Saturday, we can calculate the probability of having exactly 2, exactly 3, ..., up to exactly 15 years with New Year's Day on a Saturday, and then add up these probabilities.
Let's use the complement rule to make the calculation easier. The complement event is having 0 or 1 year with New Year's Day on a Saturday. So, we can calculate P(at least 2) by subtracting the probability of the complement event from 1.
P(at least 2) = 1 − P(0 or 1)
To calculate P(0 or 1), we can use the binomial probability formula for x = 0 and x = 1:
P(0) = (15C0) * (1/7)^0 * (6/7)^(15 − 0)
P(1) = (15C1) * (1/7)^1 * (6/7)^(15 − 1)
Then, we can calculate P(at least 2):
P(at least 2) = 1 − [P(0) + P(1)]
I hope this explanation helps you understand how to calculate the probability in each scenario based on the given information.