Question

I have a total of two bikes that I keep at work or at home. I always bike to and from my work unless it rains, in which case i take a taxi (and don't bring my bike). If the probability of rain on any leg of the commute is 2/5, what is the probability that it does not rain and I won't have a bike to use?

Answer 1

The probability that it does not rain and the student won't have a bike to use is 36/625, given the independent probability of rain and taxi use.

Step-by-step explanation:

The situation described involves probability and the probability of independent events. With two bikes and a 2/5 chance of rain for any given commute, we need to work out the possibility of being bikeless on a non-rainy day. Since there are two commutes in a day (to work and back from work), the probability of it not raining for both commutes is (1 - 2/5) * (1 - 2/5) = 3/5 * 3/5 = 9/25. Now, the only way the student won't have a bike is if it did not rain on both legs of the prior day's commute, which has a probability of 9/25, and they took a taxi both times (hence, leaving both bikes at the same location and not having a bike where needed).

To be without a bike after two taxi rides, the probability of taking a taxi in the first leg of the commute is 2/5, and it is the same for the second leg. Since the taxi rides are independent from day to day, we find the probability of no rain and having taken a taxi twice the day before: 9/25 * 2/5 * 2/5 = 36/625.

Answer 2

To calculate the probability that it does not rain and I won't have a bike to use, we can use the concept of conditional probability. Given that the probability of rain on any leg of the commute is 2/5, and the probability of having a bike to use is 2/5, the probability of it not raining and not having a bike to use is 6/25.

Step-by-step explanation:

To solve this problem, we can use the concept of conditional probability.

Let's represent the events as follows:

A: It does not rain

B: I don't have a bike to use

We want to find the probability of event (A and B), which is essentially the probability that it does not rain and I won't have a bike to use.

Given that the probability of rain on any leg of the commute is 2/5, we can determine the probability of it not raining on any given leg as 1 - 2/5 = 3/5.

Similarly, the probability of having a bike to use is 2/5.

Using these probabilities, we can calculate the probability of event (A and B) as the product of the individual probabilities: (3/5) * (2/5) = 6/25.

Therefore, the probability that it does not rain and I won't have a bike to use is 6/25.