Question

department of labor and employment (dole) found that 83% of Filipinos think that having a college education is important to succeed in life. if a random samples of seven Filipinos is selected, find this probabilities.

a. exactly four people will agree with that statement

b. at most two people will agree with that statement

c. at least five two people will agree with that statement

Answer 1

Let us consider that people of Filipino think that having college education is important in life as a success.

Then probability of success is $\frac{83}{100}$ and probability of failure is $\frac{17}{100}$.

$\therefore p=\frac{83}{100}$ ​and$q=\frac{17}{100}$.

Now we know that for binomial distribution probability of getting $r$ success from $n$ trial is $={^n}C_r.p^r.q^{(n-r)}$

(a) Here $n=7$ and $r=4$

Therefore out of seven Filipino exactly four people will agree the statement is $=^7C_4.(\frac{83}{100})^4.(\frac{17}{100})^{(7-4)}$

$=\frac{35}{(100)^7}.(83)^4.(17)^3$

(b) At most two people will agree means we have to find the probabilities for $r=0,r=1,r=2.$

Therefore at most two people will agree that statement is $=^7C_0.(\frac{83}{100})^0.(\frac{17}{100})^{(7-0)}+^7C_1.(\frac{83}{100})^1.(\frac{17}{100})^{(7-1)}+^7C_2.(\frac{83}{100})^2.(\frac{17}{100})^{(7-2)}$

$=(\frac{17}{100})^{7}+7.(\frac{83}{100}).(\frac{17}{100})^{6}+21.(\frac{83}{100})^2.(\frac{17}{100})^{5}$

(c) At least five people will agree means we have to find the probabilities for $r=5,r=6,r=7.$

Therefore at most two people will agree that statement is $=^7C_5.(\frac{83}{100})^5.(\frac{17}{100})^{(7-5)}+^7C_6.(\frac{83}{100})^6.(\frac{17}{100})^{(7-6)}+^7C_7.(\frac{83}{100})^7.(\frac{17}{100})^{(7-7)}$

$=21.(\frac{83}{100})^5.(\frac{17}{100})^{2}+7.(\frac{83}{100})^6.(\frac{17}{100})+(\frac{83}{100})^7$