Jeremy sells a magazine which is produced in order to raise money for homeless people.The probability of making a sell is independently 0.09 fr each person he approaches.Given that he approaches 40 people,find the probability that he will make;
(a) Two or few sells
solution.
B(40,0.09)
binomial, n=40P=0.09
0.0289-0.0290
(b)exactly 4 sells
solution
P(4)=0.7103-0.5092=0.2011
(c) More than 5 sells
solution
P(>5)=1-0.8535
0.146-0147=0.296
what about if he approaches 10 people?
Well, Jeremy is doing a noble thing by selling magazines to raise money for homeless people. Let's break down the probabilities for his sales:
(a) The probability of making two or fewer sales out of 40 is quite low. But hey, it's not impossible! We calculate it as 0.0289 - 0.0290. So, Jeremy has a small chance of making two or fewer sales.
(b) Now, let's focus on exactly four sales. The probability turns out to be 0.7103 - 0.5092, which is about 0.2011. So, Jeremy has a 20.11% chance of making exactly four sales.
(c) Moving on to more than five sales, we can find the probability by subtracting the probability of five or fewer sales from one. So, it's 1 - 0.8535, which is approximately 0.146 - 0.147. Therefore, Jeremy has a 14.6% chance of making more than five sales.
Remember, these probabilities are based on assuming each person he approaches is an independent event. So, let's encourage Jeremy and hope that he makes some great sales for a great cause!
To find the probability that Jeremy will make two or fewer sales, we can use the binomial distribution formula. The formula is:
P(X ≤ k) = ∑(from i=0 to k) nCi * p^i * (1-p)^(n-i)
where:
X represents the number of sales Jeremy makes
k represents the number of sales (in this case, k = 2)
n represents the number of people Jeremy approaches (n = 40)
p represents the probability of making a sale for each person (p = 0.09)
nCi represents the binomial coefficient, which can be calculated using n!/(i!(n-i)!)
Using this formula, we can calculate the probability that Jeremy will make two or fewer sales:
P(X ≤ 2) = (40C0 * 0.09^0 * (1-0.09)^(40-0)) + (40C1 * 0.09^1 * (1-0.09)^(40-1)) + (40C2 * 0.09^2 * (1-0.09)^(40-2))
This simplifies to:
P(X ≤ 2) = (1 * 1 * 0.91^40) + (40 * 0.09 * 0.91^39) + (780 * 0.09^2 * 0.91^38)
Calculating this expression will give us the probability that Jeremy will make two or fewer sales.
To find the probability of exactly 4 sales, we can use the binomial distribution formula again:
P(X = 4) = 40C4 * 0.09^4 * (1-0.09)^(40-4)
This simplifies to:
P(X = 4) = 91,390 * 0.09^4 * 0.91^36
Calculating this expression will give us the probability that Jeremy will make exactly 4 sales.
To find the probability of more than 5 sales, we can calculate the complement of the probability of 5 or fewer sales since:
P(X > 5) = 1 - P(X ≤ 5)
We have already calculated P(X ≤ 5) using the first part of this question. Subtracting it from 1 will give us the probability of more than 5 sales.
To find the probability in each case, we will use the binomial distribution formula. The formula for the binomial distribution is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the number of trials or people approached
p is the probability of success for each trial
(1-p) is the probability of failure for each trial
nCx is the binomial coefficient, which represents the number of ways to choose x successes out of n trials, and can be calculated as n! / (x!(n-x)!)
Let's solve each part step by step:
(a) Two or fewer sells:
P(0) + P(1) + P(2)
We plug in the values:
P(0) = (40C0) * 0.09^0 * (1-0.09)^(40-0)
P(1) = (40C1) * 0.09^1 * (1-0.09)^(40-1)
P(2) = (40C2) * 0.09^2 * (1-0.09)^(40-2)
We can use a binomial calculator or a statistical software to compute the combination coefficients and calculate the probabilities.
(b) Exactly 4 sells:
P(4)
Using the formula:
P(4) = (40C4) * 0.09^4 * (1-0.09)^(40-4)
(c) More than 5 sells:
P(>5) = 1 - P(0) - P(1) - P(2) - P(3) - P(4) - P(5)
We subtract the probabilities of getting 0 to 5 sells from 1 to get the probability of getting more than 5 sells.
Remember to calculate the binomial coefficients and perform the calculations to find the final probabilities.