Probability Analysis of Bob and Sarah Arriving on Time

When analyzing the probability that Bob or Sarah arrives at work on time, we must first understand the nuances of the logical #34;or#34; in probability. This article delves into the different scenarios and provides a clear explanation of the calculations.

Setting the Stage: Understanding the Probabilities

We start with the given probabilities:

The probability that Bob arrives on time is 80% (0.8). The probability that Sarah arrives on time is 90% (0.9). The probability that both arrive on time is 72% (0.72).

Based on this information, we can analyze the scenarios where Bob and Sarah are either on time, late, or a combination of the two.

Probability of Both Being On Time

The probability that both Bob and Sarah arrive on time is given as 72% (0.72). This is a key piece of information that helps us understand their timeliness independently. Therefore, the probability that at least one of them is not on time can be calculated as:

1 - 0.72 0.28 (28%)

Exclusive or (XOR) vs. Inclusive or (OR)

There are two types of #34;or#34; in logic:

Inclusive or (inclusive or): This means either one of the events or both can occur. Example: Bob is late or Sarah is late or both. Exclusive or (XOR): This means either one of the events can occur, but not both. Example: Bob is late and Sarah is on time or Bob is on time and Sarah is late.

Calculating the Probability Using Inclusive and Exclusive or

Inclusive or

Using inclusive or (OR), we calculate the probability that at least one of them is late as given above (28%). This translates to the probability that both are not on time (28%):

Exclusive or

For exclusive or (XOR), the calculation is slightly different. The probability that Bob or Sarah is late but not both can be found by first calculating the probability that both are late:

Probability that both are late: 1 - 0.8 - 0.9 0.72 0.02 (2%)

Probability of one being late and the other on time:

0.8 * 0.1 0.1 * 0.9 0.08 0.09 0.17 (17%)

Adding these probabilities together:

2% 17% 19% (26% 1 - 0.74)

Further Analysis

By establishing that the probability of Bob and Sarah both being on time (72%) assures the independence of their timeliness, we can deduce:

10% of the time, Bob is late when Sarah is on time. 20% of the time, Sarah is late when Bob is on time. Total scenarios where at least one of them is late: 28%.

Probability Combinations

The combination of these probabilities is:

BOTH IN (both on time): 72%. BOTH OUT (both late): 2%. ONE OUT (one late): 26%.

Conclusion

Given the initial probabilities, the probability that both Bob and Sarah arrive on time is 98% (0.98). Therefore, the probability that both are late is 2% (0.02), which is calculated as follows:

1 - 0.98 0.02 (2%).

This detailed analysis helps in understanding the nuances of inclusive and exclusive or in probability calculations.