Question

X and Y are two independent events. The probability that the event X will occur is twice the probability that the event Y will occur and the probability that Y will not occur is three times the probability that X will not occur. Then find the probability that both X andY will not occur.

Answer 1

Answer:

The probability that both X and Y will not occur is 0.12.

Step-by-step explanation:

It is provided that P (X) = 2 P (Y).

Also,

$P(Y^{c})=3P(X^{c})\\1-P(Y)=3[1-P(X)]\\3P(X)-P(Y)=2$

Substitute P (X) = 2 P (Y) in the above equation:

$3P(X)-P(Y)=2\\(3\times 2P(Y))-P(Y)=2\\5P(Y)=2\\P(Y)=\frac{2}{5}$

Then the value of P (X) is:

$P(X)=2P(Y)=2\times \frac{2}{5} =\frac{4}{5}$

The probability of neither X nor Y is:

$P(X^{c}\cup Y^{c})=1-P(X\cup Y)\\=1-P(X)-P(Y)+P(X\cap Y)\\=1-P(X)-P(Y)+P(X)P(Y)$

It is provided that X and Y are independent. So P (X∩Y) = P(X) P(Y).

The value of the probability of neither X nor Y is:

$P(X^{c}\cup Y^{c})=1-P(X)-P(Y)+P(X)P(Y)\\=1-\frac{4}{5}-\frac{2}{5}+ (\frac{4}{5}\times \frac{2}{5})\\=0.12$

Thus, the probability that both X and Y will not occur is 0.12.