Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 16 of the 58 boxes on the shelf have the secret decoder ring. The other 42 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?
prob of ring = 16/58
prob(2 rings in 2 draws)
= (16/58)(15/57)
= 40/551
or
C(16,2)/(C(58,2))
= 120/1653
= 40/551
Ah, the cereal box conundrum! Well, to tackle this probability problem, we need to calculate the chances of getting a box with the secret decoder ring twice in a row.
First, let's determine the probability of getting a box with the secret decoder ring on the first pick. Out of the 58 boxes total, 16 have the ring. So the probability of picking one of those is 16/58.
Now, onto the second pick. Since we haven't put the first box back on the shelf, the number of total boxes drops to 57. However, the number of boxes with the secret ring decreases to 15 because we already picked one.
So the probability of getting another box with the decoder ring on the second pick is 15/57.
To calculate the probability of both events happening, we multiply the probabilities together: (16/58) * (15/57) = 240/3294.
Simplifying the fraction, the probability of getting the secret decoder ring in both boxes is approximately 0.0729, or about 7.29%.
In conclusion, you have about a 7.29% chance of scoring two decoder rings. Just remember, even if you don't get the decoder ring, it's all in good fun. Who needs spy gadgets when you have a bowl of crunchy cereal? Enjoy your breakfast adventures!
To find the probability that both of the selected boxes have the secret decoder ring, you need to calculate the probability of selecting a box with the secret decoder ring on the first try and then multiply it by the probability of selecting another box with the secret decoder ring on the second try.
First, let's calculate the probability of selecting a box with the secret decoder ring on the first try. Out of the total 58 boxes on the shelf, 16 have the secret decoder ring. Therefore, the probability of selecting a box with the secret decoder ring on the first try is 16/58.
After selecting the first box, there will be one less box on the shelf. If the first box does have the secret decoder ring, there will be 15 remaining. If the first box does not have the secret decoder ring, there will be 16 remaining. Thus, for the second box, the probability of selecting another box with the secret decoder ring depends on the outcome of the first selection.
Assuming the first box selected does have the secret decoder ring, there will be 15 remaining boxes with the secret decoder ring out of a total of 57 remaining boxes. Therefore, the probability of selecting a box with the secret decoder ring on the second try, given that the first box has the secret decoder ring, is 15/57.
If the first box selected does not have the secret decoder ring, there will be 16 remaining boxes with the secret decoder ring out of a total of 57 remaining boxes. Therefore, the probability of selecting a box with the secret decoder ring on the second try, given that the first box does not have the secret decoder ring, is 16/57.
Now, let's calculate the overall probability by considering both scenarios:
Probability of selecting a box with the secret decoder ring on the first try: 16/58
Probability of selecting a box with the secret decoder ring on the second try, given the first box does have the secret decoder ring: 15/57
Probability of selecting a box with the secret decoder ring on the second try, given the first box does not have the secret decoder ring: 16/57
To find the probability that both boxes have the secret decoder ring, we multiply these probabilities:
(16/58) * (15/57) = 0.053
Therefore, the probability that both of the selected boxes have the secret decoder ring is approximately 0.053, or 5.3%.