We are told that two customers will be randomly selected to win a gift certificate at a clothing store and we want to know the probability that we do not both win. If we call the event of us both winning

A,

then we want to know

P (not A) .

P (A) = P (n o t A) = We both win We do n o t both win

These two probabilities are complements. The sum of the probability of an event and the probability of its complement is

1 .

P (A) + P (not A) = 1

To find the value of

P (not A),

we will calculate

P (A)

and substitute it into the above equation. We can use the theoretical probability to find

P (A) .

P (A) = \frac{Number of favorable outcomes}{Number of possible outcomes}

In this case, the number of possible outcomes is the number of possible unique pairs of winning customers. Notice that the order of selection is not important because we only care about which customers are selected. This means that we can use the formula for the number of combinations formed by groups of

r

elements from a set of

n

elements.

_{n} C_{r} = \frac{n !}{( n - r ) ! \cdot r !}

We are told that there are

19

other customers in the store with us — making the total number of customers, and the value of

n,

equal

21 .

Out of them,

2

customers will be selected to win the gift certificates, so

r

equals

2 .

_{n} C_{r} = \frac{n !}{( n - r ) ! \cdot r !}

SubstituteII

$n = 21$ , $r = 2$

_{21} C_{2} = \frac{2 1 !}{( 2 1 - 2 ) ! \cdot 2 !}

▼

Evaluate right-hand side

SubTerm

Subtract term

_{21} C_{2} = \frac{2 1 !}{1 9 ! \cdot 2 !}

Write as a product

_{21} C_{2} = \frac{2 1 \cdot 2 0 \cdot 1 9 !}{1 9 ! \cdot 2 !}

CancelCommonFac

Cancel out common factors

_{21} C_{2} = \frac{2 1 \cdot 2 0 \cdot 1 9 !}{1 9 ! \cdot 2 !}

SimpQuot

Simplify quotient

_{21} C_{2} = \frac{2 1 \cdot 2 0}{2 !}

$2! = 2$

_{21} C_{2} = \frac{2 1 \cdot 2 0}{2}

Multiply

_{21} C_{2} = \frac{4 2 0}{2}

CalcQuot

Calculate quotient

_{21} C_{2} = 210

The

number

of

possible

outcomes

is

210 .

Now, we need to find the number of

favorable

outcomes .

There is only one favorable outcome, the one in which we are both the winners of the gift certificates, so the

number

of

favorable

outcomes

is

1 .

P (A) = \frac{Number of favorable outcomes}{Number of possible outcomes} ⇓ P (A) = \frac{1}{2 1 0}

Finally, we can find the probability of the complement of

A,

P (not A),

using the value of

P (A) .

P (A) + P (not A) = 1

Substitute

$P (A) = \frac{1}{2 1 0}$

\frac{1}{2 1 0} + P (not A) = 1

▼

Solve for

P (not A)

SubEqn

$LHS - \frac{1}{2 1 0} = RHS - \frac{1}{2 1 0}$

P (not A) = 1 - \frac{1}{2 1 0}

OneToFrac

Rewrite $1$ as $\frac{2 1 0}{2 1 0}$

P (not A) = \frac{2 1 0}{2 1 0} - \frac{1}{2 1 0}

SubFrac

Subtract fractions

P (not A) = \frac{2 1 0 - 1}{2 1 0}

SubTerm

Subtract term

P (not A) = \frac{2 0 9}{2 1 0}

The probability that we will not both win the gift certificate is

\frac{2 0 9}{2 1 0} .

Exercise