Question

Dan's Diner employs three dishwashers. Al washes \(40 \%\) of the dishes and breaks only \(1 \%\) of those he handles. Betty and Chuck each wash \(30 \%\) of the dishes, and Betty breaks only \(1 \%\) of hers, but Chuck breaks \(3 \%\) of the dishes he washes. (He, of course, will need a new job soon. ....) You go to Dan's for supper one night and hear a dish break at the sink. What's the probability that Chuck is on the job?

Short answer

The probability that Chuck is on the job is 56.25%.

Step-by-step solution

Determine Each Dishwasher's Contribution to Broken Dishes

First, we determine the probability contribution from each dishwasher when a dish breaks. We need to calculate how many dishes each person is likely to break. This is done by multiplying the percentage of dishes they wash by the breakage rate.For Al: \[ P(\text{break by Al}) = 0.40 \times 0.01 = 0.004 \]For Betty: \[ P(\text{break by Betty}) = 0.30 \times 0.01 = 0.003 \]For Chuck: \[ P(\text{break by Chuck}) = 0.30 \times 0.03 = 0.009 \]

Total Probability of a Broken Dish

Add the probabilities of a dish being broken by any dishwasher to find the total probability of a dish breaking.\[ P(\text{break}) = 0.004 + 0.003 + 0.009 = 0.016 \]

Calculate the Probability Chuck Broke the Dish

Using Bayes' Theorem, we calculate the probability that Chuck broke the dish given that a dish has been broken. This is calculated as the probability Chuck broke a dish divided by the total probability of a dish breaking.\[ P(\text{Chuck}|\text{break}) = \frac{P(\text{break by Chuck})}{P(\text{break})} = \frac{0.009}{0.016} = 0.5625 \]

Convert the Probability to a Percentage

Express the result as a percentage to determine the likelihood that Chuck is responsible when a dish breaks.\[ P(\text{Chuck}|\text{break}) = 0.5625 \times 100\% = 56.25\% \]

Key concepts

Bayes' Theorem

Bayes' Theorem is a powerful tool in probability theory that helps us update beliefs with new evidence. It enables us to calculate conditional probabilities by relating the probability of an event, based on prior knowledge of conditions that might be related to the event. Here, for instance, we want to find out how likely it is that Chuck broke the dish given we know a dish broke.

• The theorem takes into account prior probabilities like how many dishes Chuck usually washes and how often he breaks them.
• We calculate the probability of each dishwasher breaking dishes by multiplying the proportion of dishes they clean with their respective breakage rates.
• Then, Bayes' Theorem combines these probabilities with the total probability of a broken dish to give us the answer.

This makes it easy to decide how probable Chuck is the culprit once we hear that loud dish-breaking sound.

Probability Calculation

Probability calculation is crucial to understanding any scenario involving uncertainty and likelihood. It involves calculating how often an outcome will occur by considering all possible outcomes. For Dan's diners' exercise:

• First, calculate each dishwasher's contribution to broken dishes based on the percentage of dishes they handle and their breakage rates. For example, Chuck washes 30% of the dishes but breaks 3% of them.
• Second, add up these probabilities to figure out the total probability of any dish breaking, which comes to 0.016 or 1.6% to be exact.

These simple arithmetic calculations lay the foundation for further inference and decision-making. Without these, determining who is most likely responsible for a broken dish wouldn't be possible.

Conditional Probability

Conditional probability is about finding the likelihood of an event occurring given that another event is already known to have happened. In this case, we've already heard a dish break, and we are trying to ascertain who is on dish duty from this information.

• We find the probability of Chuck breaking a dish, then adjust this based on the new condition — a dish has just been broken.
• This involves using Bayes’ Theorem to refine our probabilities, moving from a general case to the specific condition observed.

By applying conditional probability, we find with 56.25% certainty that Chuck broke the dish once we know a dish is indeed broken. This focused approach helps make informed decisions or predictions based on partial information and observed outcomes.