Question

nventing is a difficult way to make money. Only 5% of new patents earn a substantial profit. A certain city has just had30 independent new patents recorded.a) What is the probability that at least 2 of these new patents will earn a substantial profit?

Answer 1

`P(X \geq 2) = 1-P(X<2) = 1-P(X \leq 1) =1- [P(X=0) +P(X=1)]`

And we can find the individual probabilities using the probability mass function and we got:

`P(X=0) = (30C0) (0.05)^0 (1-0.05)^(30-0) =0.2146`

`P(X=1) = (30C1) (0.05)^1 (1-0.05)^(30-1) = 0.3389`

And replacing we got:

`P(X \geq 2) = 1-P(X<2) = 1-P(X \leq 1) =1- [P(X=0) +P(X=1)] = 1-[0.2146+0.3389] =0.4465`

Explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest, on this case we now that:

`X \sim Binom(n=30, p=0.05)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

Solution to the problem

For this case we want this probability:

`P(X \geq 2)`

And we can use the complement rule and we got:

`P(X \geq 2) = 1-P(X<2) = 1-P(X \leq 1) =1- [P(X=0) +P(X=1)]`

And we can find the individual probabilities using the probability mass function and we got:

`P(X=0) = (30C0) (0.05)^0 (1-0.05)^(30-0) =0.2146`

`P(X=1) = (30C1) (0.05)^1 (1-0.05)^(30-1) = 0.3389`

And replacing we got:

`P(X \geq 2) = 1-P(X<2) = 1-P(X \leq 1) =1- [P(X=0) +P(X=1)] = 1-[0.2146+0.3389] =0.4465`