Answer to Question #88151 in Statistics and Probability for Ritesh

Question

for two equally likely, exhaustive and independent events a and b. what is the probability of a and b

Answer 1

Suppose a and b are two events associated with a random experiment. The probability of union of two events is given by:

$$P(a\cup b)=P(a)+P(b)-P(a\cap b)$$

Equally likely events are events that have the same theoretical probability of occurring.

$$P(a)=P(b)$$

If a and b are independent events, the probability of both the events happening together

$$P(a\cap b)=P(a)P(b)$$

If a and b are exhaustive events, the probability of union of two events is

$$P(a\cup b)=1$$

Hence

$$1=P(a)+P(a)-(P(a))^2$$

$$P(a)(2-P(a))=1$$

If $P(a)=0$, then $P(b)=P(a)=0$ and $P(a\cup b)=0$. False.

Therefore, $P(a) =\not 0$ and

$$2-P(a)=1$$

$$P(a)=1=P(b)$$

$$P(a\cap b)=(1)^2=1$$