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

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

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Answer 1 · 2017-09-30T01:33:20

See below:

Explanation:

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

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

$P (A A) = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$ $P(A A)=3/4xx3/4=9/16$

$P (a a) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ $P(a a)=1/4xx1/4=1/16$

$P (A a) = \frac{3}{4} \times \frac{1}{4} = \frac{3}{16}$ $P(A a)=3/4xx1/4=3/16$

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

$P (A A) = p \times p = p^{2}$ $P(A A)=pxxp=p^2$

$P (a a) = (1 - p) \times (1 - p) = {(1 - p)}^{2} = 1 - 2 p + p^{2}$ $P(a a)=(1-p)xx(1-p)=(1-p)^2=1-2p+p^2$

$P (A a) = p \times (1 - p) = p - p^{2}$ $P(A a)=pxx(1-p)=p-p^2$

~~~~~

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

$P (A a) = 2 \times \frac{3}{4} \times \frac{1}{4} = \frac{6}{16}$ $P(A a)=2xx3/4xx1/4=6/16$

and

$P (A a) = 2 \times p \times (1 - p) = 2 p - 2 p^{2}$ $P(A a)=2xxpxx(1-p)=2p-2p^2$

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

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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: AA,aa,AaA A, aa, Aa?

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