The sum of the probability of an event and the probability of its complement is 1. Recall the formula for the number of possible combinations formed by groups of r elements taken from a set of n elements.
210209
Practice makes perfect
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(notA).
P(A)=P(notA)=WebothwinWedonotbothwin
These two probabilities are complements. The sum of the probability of an event and the probability of its complement is 1.
P(A)+P(notA)=1
To find the value of P(notA), we will calculate P(A) and substitute it into the above equation. We can use the theoretical probability to find P(A).
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.
nCr=(n−r)!⋅r!n!
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.
The numberofpossibleoutcomes is 210. Now, we need to find the number of favorableoutcomes. There is only one favorable outcome, the one in which we are both the winners of the gift certificates, so the numberoffavorableoutcomes is 1.
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