Question

According to a recent survey, 81 percent of adults in a certain state have graduated from high school. If 15 adults from the state are selected at random, what is the probability that 5 of them have not graduated from high school?

⁢(2015)(0.19)15(0.81)5
A

(105)(0.19)15(0.81)15
B

(105)(0.81)5(0.19)10
C

(155)(0.19)5(0.81)10
D

(155)(0.81)5(0.19)10
E

Answer 1

Answer:

The probability that 5 of them have  graduated from high school

             = 1 - p(X=5)

            =  $1-1001 (0.81)^{5} (0.19)^{15-5}$

         

Step-by-step explanation:

Step(i) :-

The probability of getting adults in a certain state have graduated from high school

Probability of success (p)  = 0.81

Probability of failure (q) = 1- p = 1 - 0.81 = 0.19

Given number of adults 'n' = 15

Step(ii):-

Let 'X' be a random variable in binomial distribution

$P(X=r) = n_{C_{r} } p^{r} q^{n-r}$

The probability that 5 of them have  graduated from high school

$P(X=5) = 15_{C_{5} } (0.81)^{5} (0.19)^{15-5}$

we know that

$15_{C_{5} } = \frac{15!}{(15-5)!5!} = \frac{15 X 14 X 13 X 12 X 11 X 10!}{10! 5 X4X3X2X1}$ =   1001

The probability that 5 of them have  graduated from high school

$P(X=5) = 1001 (0.81)^{5} (0.19)^{15-5}$

Step(iii):-

The probability that 5 of them have  graduated from high school

             = 1 - p(X=5)

            =  $1-1001 (0.81)^{5} (0.19)^{15-5}$

Answer 2

The probability that 5 of them are not graduated from the high school is:

$P=1-1001(0.81)^5(0.19)^{15-5}$

Step-by-step explanation:

Given information:

The probability of getting adults in a certain state have graduated from high school.

Probability of success $(p)=0.81$

Probability of failure $(q)=0.19$

Sample (number of adults) $(n)=15$

Now, in the binomial distribution

Let, $X$ be a random variable,

So,

$P(X=r) = nc_rp^rq^{n-r}$

Now , the probability that 5 of them have graduated from the high school

$P(X=5)=15c_5(0.81)^5(0.19)^{10}$

We know that :

$15c_5=1001$

On putting the values in the above equation we get:

$P(X=5)=1001(0.81)^5(0.19)^{15-5}$

Now, the probability that 5 of them are not graduated from the high school is:

$P=1-P(x=5)$

$P=1-1001(0.81)^5(0.19)^{15-5}$

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