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Answer :
To solve the problem of finding the probability that at least one of the two (either the train or the bus) arrives late, we can use the formula for the probability of the union of two events. This formula helps us find the probability of either one event or the other happening or both events happening together.
Here's the step-by-step solution:
1. Identify the Probabilities:
- The probability that the train arrives late, [tex]\( P(T) \)[/tex], is 0.2.
- The probability that the bus arrives late, [tex]\( P(B) \)[/tex], is 0.3.
- The probability that both the train and the bus arrive late, [tex]\( P(T \cap B) \)[/tex], is 0.1.
2. Use the Formula for the Union of Two Events:
- We want to find the probability that at least one arrives late, which is the probability of the union of two events, [tex]\( P(T \cup B) \)[/tex].
- The formula for this is:
[tex]\[
P(T \cup B) = P(T) + P(B) - P(T \cap B)
\][/tex]
- This formula accounts for the fact that when we add the probabilities of the train and the bus being late separately, we count the scenario where both are late twice. So, we subtract the probability of both being late once to correct for this.
3. Substitute the Known Values:
- Plug the values we identified into the formula:
[tex]\[
P(T \cup B) = 0.2 + 0.3 - 0.1
\][/tex]
4. Calculate:
- Calculate the result:
[tex]\[
P(T \cup B) = 0.4
\][/tex]
Therefore, the probability that at least one of the two modes of transport, either the train or the bus, arrives late is 0.4, or 40%.
Here's the step-by-step solution:
1. Identify the Probabilities:
- The probability that the train arrives late, [tex]\( P(T) \)[/tex], is 0.2.
- The probability that the bus arrives late, [tex]\( P(B) \)[/tex], is 0.3.
- The probability that both the train and the bus arrive late, [tex]\( P(T \cap B) \)[/tex], is 0.1.
2. Use the Formula for the Union of Two Events:
- We want to find the probability that at least one arrives late, which is the probability of the union of two events, [tex]\( P(T \cup B) \)[/tex].
- The formula for this is:
[tex]\[
P(T \cup B) = P(T) + P(B) - P(T \cap B)
\][/tex]
- This formula accounts for the fact that when we add the probabilities of the train and the bus being late separately, we count the scenario where both are late twice. So, we subtract the probability of both being late once to correct for this.
3. Substitute the Known Values:
- Plug the values we identified into the formula:
[tex]\[
P(T \cup B) = 0.2 + 0.3 - 0.1
\][/tex]
4. Calculate:
- Calculate the result:
[tex]\[
P(T \cup B) = 0.4
\][/tex]
Therefore, the probability that at least one of the two modes of transport, either the train or the bus, arrives late is 0.4, or 40%.
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