On any day when she travels to college, she uses one of three options: her bike only; a bus only; or both her bike and a bus. The probability that she uses her bike, either on its own or with the bus, is 0.7. The probability that she uses both her bike and the bus is 0.3. Calculate the probability that, on any day when she travels to college, she: a) does not use her bike; b) uses only her bike; c) uses the bus; d) uses her bike, given that she used the bus.
Answer (1)
Let's look into the problem using the information provided.
Define the events:
Let B be the event that she uses her bike.
Let Bus be the event that she uses the bus.
Given probabilities:
The probability that she uses her bike (either on its own or with the bus), P ( B ) = 0.7 .
The probability that she uses both her bike and the bus, P ( B ∩ Bus ) = 0.3 .
Using this information, we can compute the required probabilities step by step.
a) Probability that she does not use her bike:
The probability of not using her bike is the complement of the probability of using her bike.
P ( Not B ) = 1 − P ( B ) = 1 − 0.7 = 0.3
b) Probability that she uses only her bike:
To find this probability, we subtract the probability that she uses both the bike and the bus from the probability that she uses her bike.
P ( B only ) = P ( B ) − P ( B ∩ Bus ) = 0.7 − 0.3 = 0.4
c) Probability that she uses the bus:
We use the formula for the probability of a union of events to find the probability that she uses the bus.
P ( Bus ) = P ( B ∪ Bus ) = P ( B ) + P ( Bus ) − P ( B ∩ Bus )
Since we want P ( Bus ) and we don't know P ( B ∪ Bus ) , we rearrange it:
Using: P ( B ∪ Bus ) = 1
Therefore, solving for P ( Bus ) :
P ( Bus ) = 1 − P ( B ) + P ( B ∩ Bus )
P ( Bus ) = 1 − 0.7 + 0.3 = 0.6
d) Probability that she uses her bike, given that she used the bus:
This is conditional probability, given by:
P ( B ∣ Bus ) = P ( Bus ) P ( B ∩ Bus )
Substitute the known values:
P ( B ∣ Bus ) = 0.6 0.3 = 0.5
Therefore, the probabilities are as follows:
a) The probability that she does not use her bike is 0.3.
b) The probability that she uses only her bike is 0.4.
c) The probability that she uses the bus is 0.6.
d) The probability that she uses her bike, given that she used the bus, is 0.5.
