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If the probability of death from a risky activity is 1% If you carry out that activity 200 times, what is the probability of death?

 

in Data Science by Diamond (67,406 points)
edited by | 284 views
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Hi

The solution given below is not answering the question that Prof A G Mutambara paused. The instant you say the probability of death each time you do the trial you are answering a different question in that the probability of death each time you engage in the activity is already given, it remains as 0.01 meaning that the probability of surviving is 0.99.

What Prof Mutambara is asking is: Given that the probability of dying during an activity is 0.01 what are the odds (the probability) that you might die if you engage in the activity 200 times. The question is not about the probability of dying each time you engage in the activity but one of exposure. For an insurer, the premium of a person who drives at high speed every time they are on the road is higher than that of a person who drives below the speed limit for instance because the exposure to risk of the speedster is higher. The binomial model is the correct model to apply here.

However it is not a straightforward Binomial model application in that you do not die more than once so x = 0 all the time what varies is n the number of trials or number of exposures! And the higher the number of exposures the higher the chance that you may die during the process.

Your definition of 0.99 is not the same as in Prof Mutambara's question. If 0.01 is the probability of dying then 0.99 is the probability of not dying.
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I hear what you saying but the wording of the problem implies conditional probability because the 0.01 given is a probability of death on condition that it was as a result of risky activity.

1 Answer

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Best answer

It is tempting to treat this like it is a Bernoulli experiment repeated 200 times resulting in a Binomial distribution as given below:

Answer:

$$1- (0.99)^{200} = 86.7\%$$

You are almost guaranteed of DEATH. Be careful about the risky things we do in life, including driving at high speed.

I disagree with this answer by Prof Arthur Mutambara  on his LinkedIn. Here is my thinking.

1. The question requires the use of conditional problem given the way it is worded. Without loss of meaning, the same question can be written as follows:

Given that it is a risky activity, the probability of death is $1\%$. If you carry out that activity 200 times, what is the probability of death?

Using Bayes Theorem:

$$P(D|R) = \dfrac{P(R|D) \times P(D)}{P(R)}$$

Where

$P(D)$ is the probability of death each time (what we are trying to establish)

$P(R)$ is the risk rate (e.g. associated risk of being on the road)

$P(D|R)$ is the probability of death given that it was a risky activity

$P(R|D) is the risk  given death

From the question $P(D|R) = 0.01$

From the UNECA website $P(R|D) =\dfrac{26.6}{100000}$ in Africa.

From intuition, $P(R)=0.99$ implying that the risk on the roads is high.

Substituting in Bayes Theorem:

$$0.01 = \dfrac{\dfrac{26.6}{100000} \times P(D)}{0.99}$$

Therefore $P(D) = 37.22\%$ everytime someone engages with the risky activity of travelling on some African road.

The 200 events are independent thus the only metric that matters is the probability of each individual outing.

 

 

 

by Diamond (67,406 points)
edited by

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