Question

every cereal box has a gift inside, but you cannot tell from the outside what the gift is. the store manager assures you that 19 of the 54 boxes on the shelf have the secret decoder ring. the other 35 boxes on the shelf have a different gift inside. if you randomly select two boxes of cereal from the shelf to purchase, what is the probability that both of them have the secret decoder ring? (give answer as a decimal correct to four decimal places.)

Explanation:

Step1: Calculate probability of first box

The probability of selecting a box with the secret decoder ring on the first - pick is the number of boxes with the secret decoder ring divided by the total number of boxes.
$P_1=\frac{19}{54}$

Step2: Calculate probability of second box

After picking one box with the secret decoder ring, there are 18 boxes with the secret decoder ring left and 53 boxes in total left. So the probability of selecting a box with the secret decoder ring on the second - pick given that the first box had the secret decoder ring is $P_2=\frac{18}{53}$.

Step3: Calculate the probability of both events

Since the two events are independent in the sense of conditional probability for sequential selection, the probability that both boxes have the secret decoder ring is the product of the probabilities of each event.
$P = P_1\times P_2=\frac{19}{54}\times\frac{18}{53}=\frac{19\times18}{54\times53}=\frac{342}{2862}\approx0.1195$

Answer:

$0.1195$