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#