Jason goes to school either by bus or on foot. If he gets up late in the morning, the probability that he goes by bus is 4/5. If he does not get up late in the morning, the probability that he goes by bus is 1/10. If the probability that he gets up late in morning is 1/20, find the probability that Jason goes to school on foot in a school day.
Question
Jason goes to school either by bus or on foot. If he gets up late in the morning, the probability that he goes by bus is 4/5. If he does not get up late in the morning, the probability that he goes by bus is 1/10. If the probability that he gets up late in morning is 1/20, find the probability that Jason goes to school on foot in a school day.
Solution
To solve this problem, we need to use the law of total probability.
Step 1: Identify the probabilities given in the problem.
- The probability that Jason gets up late (L) is 1/20.
- The probability that he goes by bus when he gets up late (B|L) is 4/5.
- The probability that he does not get up late (~L) is 1 - 1/20 = 19/20.
- The probability that he goes by bus when he does not get up late (B|~L) is 1/10.
Step 2: Calculate the total probability that Jason goes by bus (B).
P(B) = P(B and L) + P(B and ~L) = P(B|L)P(L) + P(B|~L)P(~L) = (4/5)(1/20) + (1/10)(19/20) = 4/100 + 19/100 = 23/100.
Step 3: Since the only two ways Jason can go to school are by bus or on foot, the probability that he goes on foot (F) is 1 - P(B).
P(F) = 1 - P(B) = 1 - 23/100 = 77/100.
So, the probability that Jason goes to school on foot in a school day is 77/100.
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