It is known that 77% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 62 new products, find the following probabilities. (Round your answers to four decimal places.)
(d) within 2 years fewer than 10 succeed
When I computed this, it came out very oddly.. The C is throwing me off. cNr?
Yes
C(n,r) = nCr
= n!/(r!(n-r)!)
e.g. C(62,2) = 62!/(2!60!) = 1891
most calculators have the nCr key, mine is the 2nd function of the 5 key
e.g. to find C(62,2)
enter:
62
nCr
2
=
to get 1891
PS. I did not take the time to find the final answer to the above string of probabilities.
To find the probability that within 2 years fewer than 10 new products succeed, we first need to determine the number of successes out of 62 new products based on the given failure rate.
The failure rate is given as 77%, which means 77% of the new products introduced fail. Therefore, the success rate is 100% - 77% = 23%.
To find the probability of fewer than 10 successes, we can sum up the probabilities of getting 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 successes out of 62 new products.
The probability of getting exactly k successes out of n trials can be calculated using the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- n is the total number of trials (62 in this case),
- k is the desired number of successes (less than 10 in this case),
- p is the probability of success in a single trial (23% or 0.23 in this case), and
- C(n, k) is the number of combinations of n items taken k at a time, which can be calculated as n! / (k! * (n - k)!).
Using this formula, we can calculate the probabilities for each value of k from 0 to 9 and sum them up to get the final probability.
P(X < 10) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 9)
Calculating each individual probability:
P(X = 0) = C(62, 0) * (0.23)^0 * (1 - 0.23)^(62 - 0)
P(X = 1) = C(62, 1) * (0.23)^1 * (1 - 0.23)^(62 - 1)
P(X = 2) = C(62, 2) * (0.23)^2 * (1 - 0.23)^(62 - 2)
.
.
.
P(X = 9) = C(62, 9) * (0.23)^9 * (1 - 0.23)^(62 - 9)
Finally, sum up all these probabilities to get P(X < 10).
P(X < 10) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 9)
Note: Calculations can be time-consuming, so it's recommended to use a calculator or statistical software to compute these probabilities.
prob(fewer than 10 succeed) --> prob(none succeeds + one succeeds + 2 succeed + ...+ 9 succeed)
= (.77)^62 + C(62,1)(.23)(.77^61) + C(62,2)(.23^2)(.77^60) + ... + C(62,9)(.23^9)(.77^53)
= ....
lots of button-pushing here