Stats: Help With Multiplication Rule

• shawnz1102
In summary, the conversation discusses the difficulty of applying multiplication rules to a problem involving independent and dependent events. The problem involves finding the probability of a driver having an accident, given their risk category and the accident rates for each category. Ultimately, the solution involves calculating a weighted average of the accident rates for each group.
shawnz1102
I am having a difficult time grasping the multiplication rules - The rules are easy to remember but actually applying it to a problem is so confusing and frustrating as I don't know whether to use Rule #1 (independent events) or Rule #2 (Dependent events).

Here's a problem from my book that I just cannot solve:

An insurance company classifies drivers as low-risk, medium-risk, and high-risk. Of those insured, 60% are low risk, 30% are medium risk, and 10% are high risk. After a study, the company finds that during a 1-year period, 1% of the low risk drivers had an accident, 5% of the medium risk drivers had an accident, and 09% of the high risk drivers had an accident. If a drier is selected at random, find the probability that the driver will have had an accident during that year.

Ans: 0.03

What i tried doing was (0.01/0.6 * 0.05/0.3 * 0.09/0.1); looks like that's not the answer :(

Does anyone know how to solve this, and whether it's an independent or dependent event?

shawnz1102 said:
I am having a difficult time grasping the multiplication rules - The rules are easy to remember but actually applying it to a problem is so confusing and frustrating as I don't know whether to use Rule #1 (independent events) or Rule #2 (Dependent events).

Here's a problem from my book that I just cannot solve:

What i tried doing was (0.01/0.6 * 0.05/0.3 * 0.09/0.1); looks like that's not the answer :(

Does anyone know how to solve this, and whether it's an independent or dependent event?

The expression you show above is wrong on two counts - you shouldn't be dividing by the group percentages, and you shouldn't be multiplying the three fractions.

This probability will be a weighted average of the accident rates in the three groups.
The probability is (accident rate in low-risk group * relative proportion of low-risk group) + (accident rate in medium-risk group * relative proportion of medium-risk group) + (accident rate in high-risk group * relative proportion of high-risk group)

Thank you so much Mark!

When I encounter a problem like this, how do i know that it will be a weighted average problem?

I wouldn't characterize it as a "weighted average problem"; I was just describing the expression I got. It's basically a probability problem with three mutually exclusive groups: low-risk, medium-risk, and high-risk drivers, and the relative probabilities of each group.

Got it, thanks!

1. How can I use the multiplication rule to solve probability problems?

The multiplication rule is a fundamental concept in probability that states that the probability of two independent events occurring together is equal to the product of their individual probabilities. To use this rule, you first need to identify the two events and determine if they are independent. Then, you can simply multiply the probabilities of each event to find the probability of them both occurring together.

2. What is the difference between the multiplication rule and the addition rule in statistics?

The multiplication rule and the addition rule are both used to calculate probabilities in statistics, but they are applied in different situations. The multiplication rule is used when calculating the probability of two independent events occurring together, while the addition rule is used when calculating the probability of either one of two mutually exclusive events occurring. In other words, the multiplication rule is used for "and" scenarios, while the addition rule is used for "or" scenarios.

3. Can the multiplication rule be applied to more than two events?

Yes, the multiplication rule can be extended to more than two events. For three independent events A, B, and C, the probability of all three events occurring together would be P(A) x P(B) x P(C). This can be extended to any number of independent events, as long as they are truly independent and not affected by each other.

4. How do I know if two events are independent or not?

Two events are considered independent if the outcome of one event does not affect the outcome of the other. In other words, the probability of one event occurring is not influenced by the occurrence or non-occurrence of the other event. To determine if two events are independent, you can use the rule P(A and B) = P(A) x P(B). If this equation holds true, then the events are independent.

5. Can the multiplication rule be applied to dependent events?

No, the multiplication rule can only be applied to independent events. For dependent events, a different rule called the conditional probability rule is used. This rule takes into account the dependence between the events and adjusts the probabilities accordingly. It is important to properly identify whether events are independent or dependent before applying any probability rules.

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