Final answer:
The probability of selecting an apple from the box is 30/143. This is found by accounting for the given probabilities and ratios of the other fruit types and then subtracting from the total probability of 1.
Step-by-step explanation:
The question asks us to determine the probability that a randomly selected piece of dried fruit from a box is an apple, given that the probability of it being a mango is 3/13 and the ratio of dried mango to dried pineapple is 3:8.
First, let's establish the ratio of mangoes to pineapples:
- Mangoes: 3 parts
- Pineapples: 8 parts
Since the probability of picking a mango is 3/13, this means that out of every 13 pieces of fruit, 3 are mangoes. Combining the ratio of mangoes to pineapples, we can infer that the total parts representing mangoes and pineapples together is 3 + 8 = 11 parts.
However, the total probability must add up to 1 (or 13/13). We currently have 3/13 for mangoes, which leaves us with 10/13 for pineapples and apples together.
As the mango to pineapple ratio is 3:8, the probability of pineapples is 8/11 of the remaining 10/13 probability. To find this, we multiply (8/11) × (10/13) which simplifies to 80/143.
Now subtract the combined probability of picking a mango or pineapple from the total probability to find the probability of picking an apple.
Total probability - Probability of mango or pineapple = Probability of apple
1 - (3/13 + 80/143) = Probability of apple
To make the subtraction easier, convert 3/13 to have a denominator of 143: (3/13) × (11/11) = 33/143.
Now subtract: 143/143 - (33/143 + 80/143) which equals 143/143 - 113/143 = 30/143. This is the probability of picking an apple in its simplest form.