A toy manufacturing company has categorised all the toys they make into 4 categories depending on the battery life of toys (assuming all the toys they manufacture run on battery). It is assumed that the toys belonging to category 1 will last for more than 50 hours with probability 0.4, while the corresponding probabilities for toys belonging to categories 2, 3, and 4 are 0.5, 0.8, and 0.2 respectively. Suppose that 30% of the toys are in category 1, 20% in category 2, 10% in category 3, and 40% in category 4. Sarah went to the toy manufacturing company and randomly picked one toy. If the toy did not last for more than 50 hours, what is the probability that she picked the toy from category 4 (assuming the toy was in use until the battery life got over)? Enter the answer upto 3 decimal accuracy
Answers
Given : A toy manufacturing company has categorized all the toys they make into 4 categories depending on the battery life of toys
category 1 will last for more than 50 hours with probability 0.4, corresponding probabilities for toys belonging to categories 2, 3, and 4 are 0.5, 0.8, and 0.2 respectively.
30% of the toys are in category 1, 20% in category 2, 10% in category 3, and 40% in category 4. Sarah went to the toy manufacturing company and randomly picked one toy
the toy did not last for more than 50 hours,
To Find : what is the probability that she picked the toy from category 4
Solution:
Assume Total Toys = 100T
30% of the toys are in category 1 = 30T
20% of the toys are in category 2 = 20T
10% of the toys are in category 3 = 10T
40% of the toys are in category 4 = 40T
category 1 will last for more than 50 hours with probability 0.4 = 0.4 * 30T = 12T
category 1 will not last for more than 50 hours = 30T - 12T = 18T
category 2 will last for more than 50 hours with probability 0.5 = 0.5 * 20T = 10T
category 2 will not last for more than 50 hours = 20T - 10T = 10T
category 3 will last for more than 50 hours with probability 0.8 = 0.8 * 10T = 8T
category 3 will not last for more than 50 hours = 10T - 8T = 2T
category 4 will last for more than 50 hours with probability 0.2 = 0.2 * 40T = 8T
category 4 will not last for more than 50 hours = 40T - 8T = 32T
the toy did not last for more than 50 hours = 18T + 10T + 2T + 32T = 62T
probability that she picked the toy from category 4 = 32T/62T
= 32/62
= 16/31
= 0.516
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