The probability of event $A$ occurring is $p$ . If $A$ doesn't occur, that is event $a$ . What is the probability of the following occurring: $A A, aa, Aa$ ?

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Answer 1 · 2017-09-30T01:33:20

See below:

Let's use some numbers first and then generalize. I'll set $A$ as probability $3/4$ and $a$ as probability $1/4$ .

The probability of drawing the three different draws are (and I'm assuming order matters and so $P(A a) and P(a A)$ are different):

$P(A A)=3/4xx3/4=9/16$

$P(a a)=1/4xx1/4=1/16$

$P(A a)=3/4xx1/4=3/16$

We can now generalize using $p$ and $1-p$ :

$P(A A)=pxxp=p^2$

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

$P(A a)=pxx(1-p)=p-p^2$

~~~~~

If order doesn't matter, then we multiply the $P(A a)$ results by 2 to account for $P(a A)$ , giving:

$P(A a)=2xx3/4xx1/4=6/16$

and

$P(A a)=2xxpxx(1-p)=2p-2p^2$

The probability of event AA occurring is pp. If AA doesn't occur, that is event aa. What is the probability of the following occurring: A A, aa, AaA A, aa, Aa?