Question

Questions pointA retailer sends scratch-off coupons to registered customers; 30% of the coupons will reveal a discount of 50%.Before mailing, a manager selects 10 coupons at random from the stack of printed coupons and scratches toreveal the discountWhat is the probability that at least one of the coupons will reveal a discount of 50%?Round to 3 decimal places,

Answer 1

The binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times.

In this case since each scratch cupon can be seen as an experiment with two possible outcomes we can use the binomial distribution.

The probability in a binomial experiment is given as:

`P(X=k)=(n!)/(k!(n-k)!)p^k(1-p)^(n-k)`

where k is the number of successes we want, n is the number of experiments we make and p is the probability of success.

In this case we want the proability that at least one cupon has a discount, that is, we want the probability:

`P(X\ge1)`

To find this probability we can add the probability of obtaining one plus the proabbility of obtaining two plus the proabibility of obtaining three and so on until we reach the proability of obtaining 10. The other way to obtain it is by using the fact that the sum of proability of all possible outcomes is equal to 1, then we can use:

`P(X\ge1)=1-P(X=0)`

In this case we only need to evaluate one probability instead of 9. For the probability P(X=0) in our case we have that n=10 (we have ten cupons), k=0 (we want the proability of obtaining zero cupons with discount) and p=0.3 (the proability of obtaining a cupon with discount is 30%), then:

`\begin{gathered} P(X\ge1)=1-(10!)/(0!(10-0)!)(0.3)^0(1-0.3)^(10-k) \\ =0.972 \end{gathered}`

Therefore the probability of obtaining at least one cupon with 50% discount is 0.972