Question

Seventy percent of kids who visit a doctor have a fever, and \(30 \%\) of kids with a fever have sore throats. What's the probability that a kid who goes to the doctor has a fever and a sore throat?

Short answer

The probability is 21%.

Step-by-step solution

Understanding the Problem

We need to determine the probability that a kid who visits the doctor has both a fever and a sore throat. We know that 70% of kids who visit a doctor have a fever. Out of those, 30% have a sore throat.

Finding the Probability of Fever

Let \( P(F) \) represent the probability of a kid having a fever. Given: \( P(F) = 70\% = 0.70 \).

Finding the Probability of Sore Throat Given Fever

Let \( P(S|F) \) represent the probability of having a sore throat given that the kid has a fever. Given: \( P(S|F) = 30\% = 0.30 \).

Applying the Multiplication Rule of Probability

We want the probability that a kid has both a fever and a sore throat. According to the multiplication rule of probability, \( P(F \text{ and } S) = P(F) \times P(S|F) \).

Calculating the Combined Probability

Substitute the known probabilities: \( P(F \text{ and } S) = 0.70 \times 0.30 = 0.21 \).

Finalizing the Answer

Convert the probability into a percentage: \( 0.21 \) is equivalent to \( 21\% \). Thus, the probability that a kid who goes to the doctor has both a fever and a sore throat is 21%.

Key concepts

Multiplication Rule of Probability

The multiplication rule of probability is a fundamental concept that helps us understand the probability of two events happening together. It is especially useful when dealing with conditional probabilities.
Indeed, when you have two events, like having a fever and a sore throat, the multiplication rule allows you to calculate the probability of both events happening at the same time.
To apply this rule, you multiply the probability of the first event by the probability of the second event occurring given that the first event has occurred. Mathematically, it is expressed as:

• \( P(A \text{ and } B) = P(A) \times P(B|A) \)

Here, \( P(A) \) is the probability of event A, and \( P(B|A) \) is the probability of event B occurring after A has happened. So, in our example:

• Event A is kids having a fever.
• Event B is kids having a sore throat, given they have a fever.

Probability Calculation

Probability calculation involves determining the likelihood of an event happening. It ranges from 0 (impossible) to 1 (certain) and can be expressed as a percentage by multiplying by 100.
In many problems, such as the one we're examining, you'll often deal with conditional probabilities. Conditional probability asks, "what's the likelihood of one event, given another event has already occurred?" This is denoted \(P(B|A)\).
In the context of the given exercise, we're looking at how likely it is for a child to experience both a fever and a sore throat. To do this, it's important to first establish:

• The probability of having a fever, \(P(F)\), which is 0.70 or 70%.
• The probability of having a sore throat given the child has a fever, \(P(S|F)\), which stands at 0.30 or 30%.

Multiplying these two probabilities gives us the joint probability we're looking to find.

Probability of Joint Events

The probability of joint events is about calculating the chance that multiple events occur together. This occurs frequently in real-world situations, where outcomes are not isolated but interconnected.
For our exercise, we want to find out the probability of a child who visits the doctor having both a fever and a sore throat simultaneously.
To find this combined probability, apply the multiplication rule:

• We calculated \( P(F \text{ and } S) \) by multiplying \( P(F) = 0.70 \) and \( P(S|F) = 0.30 \).

This results in a joint probability of 0.21. This tells us that 21% of children visiting the doctor are likely to have both conditions.
Joint probabilities can often reveal connections between events that aren't immediately obvious, offering valuable insight into how events interact.