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Chapter 14: Problem 23

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

Expert verified
The probability is 21%.

Step by step solution

01

Understanding the Problem

We need to find the probability that a kid visiting the doctor has both a fever and a sore throat. We know that 70% of kids visiting have a fever, and 30% of those with a fever have a sore throat.
02

Identifying Relevant Probabilities

Let \( P(F) \) be the probability of having a fever, which is 0.7, and \( P(S|F) \) be the probability of having a sore throat given that the kid has a fever, which is 0.3.
03

Use the Multiplication Rule for Probability

The probability that a kid has both a fever and sore throat, \( P(F \cap S) \), is given by \( P(F) \times P(S|F) \). This is because \( P(S|F) \) is the probability of having a sore throat given the kid already has a fever.
04

Calculating the Probability

Calculate \( P(F \cap S) \) by multiplying the probabilities: \( P(F \cap S) = 0.7 \times 0.3 = 0.21 \).
05

Conclusion

The probability that a kid visiting the doctor has both a fever and a sore throat is 0.21 or 21%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability is an essential concept in probability and statistics. It refers to the probability of an event occurring given that another event has already occurred. For example, in our exercise, we want to find the probability of a child having a sore throat, assuming they already have a fever.

To denote conditional probability, we often use the symbol \( P(A|B) \), which translates to the probability of event \( A \) occurring given that event \( B \) has occurred. In our scenario:
  • \( P(S|F) = 0.3 \): This means there is a 30% probability of having a sore throat given that a kid has a fever.
Understanding conditional probability helps in assessing risk or likelihood in situations where outcomes are dependent on previous events.
Multiplication Rule
The multiplication rule is a fundamental principle in probability that helps us calculate the probability of two events happening together, known as joint probability. This rule is particularly useful when dealing with sequential or dependent events, such as our exercise.

The rule states that the probability of both events \( A \) and \( B \) occurring is equal to the probability of event \( A \) happening multiplied by the probability of event \( B \) happening given \( A \) has occurred.
  • Mathematically, this is expressed as: \( P(A \cap B) = P(A) \times P(B|A) \).
  • In our case: \( P(F \cap S) = P(F) \times P(S|F) \).
  • This translates to: \( P(F \cap S) = 0.7 \times 0.3 = 0.21 \).
By applying the multiplication rule, we calculated the probability that a child has both a fever and a sore throat to be 0.21, or 21%.
Probability Calculation
Calculating probability involves determining how likely an event is to happen. In our example, we used the conditional probability and multiplication rule to find the probability of two related events happening together.

When calculating probabilities:
  • Identify all relevant probabilities, such as \( P(F) \) and \( P(S|F) \) in our exercise.
  • Use appropriate rules, like the multiplication rule, to combine these probabilities.
  • Perform the necessary arithmetic calculations, such as multiplying 0.7 by 0.3 to get 0.21.
Accurate probability calculations provide valuable insights into the likelihood of complex scenarios, thus aiding in better decision-making. In any probability calculation, always ensure to interpret the result correctly in the context of the problem to draw meaningful conclusions.

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Most popular questions from this chapter

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?

You are dealt a hand of three cards, one at a time. Find the probability of each of the following. a) The first heart you get is the third card dealt. b) Your cards are all red (that is, all diamonds or hearts). c) You get no spades. d) You have at least one ace.

After surveying 995 adults, \(81.5 \%\) of whom were over \(30,\) the National Sleep Foundation reported that \(36.8 \%\) of all the adults snored. \(32 \%\) of the respondents were snorers over the age of \(30 .\) a) What percent of the respondents were under 30 and did not snore? b) Is snoring independent of age? Explain.

If the sex of a child is independent of all other births, is the probability of a woman giving birth to a girl after having four boys greater than it was on her first birth? Explain.

In its monthly report, the local animal shelter states that it currently has 24 dogs and 18 cats available for adoption. Eight of the dogs and 6 of the cats are male. Find each of the following conditional probabilities if an animal is selected at random: a) The pet is male, given that it is a cat. b) The pet is a cat, given that it is female. c) The pet is female, given that it is a dog.
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