S1 probability problem

please could someone aid me in answering the following question.

Ahmed goes to school either by car or bicycle. If it is raining at 7.30am in the morning, the probability of Ahmed going to school by car is 0.7. If it is not raining in the morning, the probability that he goes to school by car is 0.4. The probability of rain at 7.30 in the morning is 0.1. A day is selected at random.

a) find the probability that Ahmed cycles to school.

Ahmed's teacher sees him cycling to school.
b) find the probability that is was raining at 7.30 that morning.

I have found a) to be 0.57 however i am unsure about part b)

Please help TSR

Reply 1

James

14

Partition Theorem:
P(cycles) = P(cycles | rain)P(rain) + P(cycles | dry)P(dry)

Bayes' Theorem:
P(rain | cycles) = P(cycles | rain)P(rain) / P(cycles)

Reply 2

foxygreg

1

well a) is simple if you draw a tree diagram
b) is just finding the probability that he cycles to school and it was raining divided by the probability you got from the first question but really both r simple if you draw a tree diagram and then think a little intuitivly or alternativly do a venn diagram but the question is really calling out for a tree diagram seeing as it gives conditional probability.

Reply 3

James

14

Yes, for simple problems like this (where there are a small number of partitions and straightforward dependence) a tree diagram is always a great place to start. It is useful to learn the formulae for conditional probability etc though, as sometimes it is impractical to visualise the problem in a diagram.

Reply 4

Excalibur

15

gibsion

please could someone aid me in answering the following question.

Ahmed goes to school either by car or bicycle. If it is raining at 7.30am in the morning, the probability of Ahmed going to school by car is 0.7. If it is not raining in the morning, the probability that he goes to school by car is 0.4. The probability of rain at 7.30 in the morning is 0.1. A day is selected at random.

a) find the probability that Ahmed cycles to school.

Ahmed's teacher sees him cycling to school.
b) find the probability that is was raining at 7.30 that morning.

I have found a) to be 0.57 however i am unsure about part b)

Please help TSR

To echo the others, first draw a tree diagram: the first branches will divide into R and R', rain and not rain, with probabilities of 0.1 and 0.9 respectively. Then the branches will further divide into say C and B (for car and bicycle) for each R and R'. Put down the respective conditional branches.

a) From this it becomes quite simple. The probability that he cycles to school is P(RnB) + P(R'nB) (ie. he cycles and it's raining, plus he cycles and it's not raining). This will be (0.1x0.3) + (0.9x0.6) = 0.57

b) This is the probability of it was raining, given that he went by bike. So in notation it will be P(R | B). By definition P(R | B) = P(RnB) / P(B). P(RnB) is 0.1 x 0.3 = 0.03 and P(B) you find in part b) to be 0.57. So answer is 0.03/0.57 = 1/19

S1 probability problem

Related discussions

Articles for you