Inside the box of every cereal box contains a surprise gift. The store manager assures employees that 20 of the 47 boxes on the shelf have a secret decoder ring and inside the other 27 boxes on the shelf contain a different gift. If two cereal boxes are randomly selected 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 least three decimal places.)
Question
Inside the box of every cereal box contains a surprise gift. The store manager assures employees that 20 of the 47 boxes on the shelf have a secret decoder ring and inside the other 27 boxes on the shelf contain a different gift. If two cereal boxes are randomly selected 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 least three decimal places.)
Solution
To solve this problem, we need to use the concept of probability.
Step 1: Identify the total number of outcomes. In this case, the total number of cereal boxes on the shelf is 47.
Step 2: Identify the number of favorable outcomes. In this case, the number of boxes with a secret decoder ring is 20.
Step 3: Calculate the probability of the first box having a secret decoder ring. This is done by dividing the number of favorable outcomes by the total number of outcomes. So, the probability is 20/47 = 0.4255 (rounded to four decimal places).
Step 4: Now, if one box has already been taken, there are only 46 boxes left on the shelf, and only 19 of them have a secret decoder ring. So, the probability of the second box also having a secret decoder ring is 19/46 = 0.4130 (rounded to four decimal places).
Step 5: Since we are looking for the probability of both events happening (i.e., both boxes having a secret decoder ring), we need to multiply the probabilities calculated in steps 3 and 4. So, the final probability is 0.4255 * 0.4130 = 0.1756 (rounded to four decimal places).
So, the probability that both cereal boxes selected randomly from the shelf have a secret decoder ring is approximately 0.176 (rounded to three decimal places).
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