Question

Police in a small town plan to enforce speed limits by installing'radar traps'at the two main town entrances: east end and west end. Traffic statistics show that $40 \%$ of the cars entering the town use the east entrance and the rest use the west one. The east entrance traps are operated $40 \%$ of the time and the west entrance are operated $60 \%$ of the time. Assuming that the proportion of'speeders'is the same at both entrances, what is the probability that: a) a speeding driver is spotted passing one of the traps? b) a speeding driver who got spotted has actually come from the west end?

Name: Problem 3, Chapter 12: Higher Level Mathematics
Uploaded: 2020-05-21T17:58:02-08:00
Duration: 1 min 30 s
Channel: Julian Wong

   Police in a small town plan to enforce speed limits by installing'radar traps'at the two main town entrances: east end and west end. Traffic statistics show that $40 \%$ of the cars entering the town use the east entrance and the rest use the west one. The east entrance traps are operated $40 \%$ of the time and the west entrance are operated $60 \%$ of the time. Assuming that the proportion of'speeders'is the same at both entrances, what is the probability that:
a) a speeding driver is spotted passing one of the traps?
b) a speeding driver who got spotted has actually come from the west end?

Higher Level Mathematics

Ibrahim Wazir, Tim… 2012 Edition

Chapter 12, Problem 3

Instant Answer

Solved by Expert Julian Wong

05/21/2020

Step 1

This can be calculated by adding the probabilities of a car being trapped at the east entrance and the west entrance. The probability of a car being trapped at the east entrance is the product of the probability of a car entering from the east entrance and the Show more…

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Police in a small town plan to enforce speed limits by installing'radar traps'at the two main town entrances: east end and west end. Traffic statistics show that $40 \%$ of the cars entering the town use the east entrance and the rest use the west one. The east entrance traps are operated $40 \%$ of the time and the west entrance are operated $60 \%$ of the time. Assuming that the proportion of'speeders'is the same at both entrances, what is the probability that: a) a speeding driver is spotted passing one of the traps? b) a speeding driver who got spotted has actually come from the west end?

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Key Concepts

Law of Total Probability

The law of total probability allows the calculation of the overall probability of an event by taking into account all different ways that event can occur, weighted by the probability of each way. In this case, it is used to combine the probabilities from both the east and west entrances.

Bayes' Theorem

Bayes' theorem is a method for updating probabilities based on new data. It relates the conditional probability of observing a specific cause given an effect to the probability of that effect when the cause is known. This concept is critical for determining the likelihood that a speeding driver came from the west entrance given that they were caught.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. In problems like this, it is used to compute the probability of a speeding driver being caught by a trap given that they used a particular town entrance.

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