Questions: Ive the problem that involves probabilities with events that are not mutually exclusive. There are 37 chocolates in a box, all identically shaped. There are 16 filled with nuts, 11 with caramel, and 10 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting 2 solid chocolates in a row. Answer as a reduced fraction.

Solution Steps

To find the probability of selecting 2 solid chocolates in a row, we need to consider the probability of selecting a solid chocolate on the first draw and then the probability of selecting another solid chocolate on the second draw, given that one solid chocolate has already been removed.

1. Calculate the probability of selecting a solid chocolate on the first draw.
2. Calculate the probability of selecting a solid chocolate on the second draw, given that one solid chocolate has already been removed.
3. Multiply these probabilities to get the final answer.

The probability of selecting a solid chocolate on the first draw is given by:

\[ P(\text{first solid}) = \frac{\text{number of solid chocolates}}{\text{total chocolates}} = \frac{10}{37} \]

After selecting one solid chocolate, there are now 9 solid chocolates left and a total of 36 chocolates remaining. The probability of selecting a solid chocolate on the second draw is:

\[ P(\text{second solid} | \text{first solid}) = \frac{\text{remaining solid chocolates}}{\text{remaining total chocolates}} = \frac{9}{36} = \frac{1}{4} \]

The total probability of selecting two solid chocolates in a row is the product of the probabilities from Step 1 and Step 2:

\[ P(\text{two solids}) = P(\text{first solid}) \times P(\text{second solid} | \text{first solid}) = \frac{10}{37} \times \frac{1}{4} = \frac{10}{148} = \frac{5}{74} \]

Final Answer