Question

Current estimates are that about 25% of all deaths are due to cancer, and of the deaths that are due to cancer, 30% are attributed to tobacco, 40% to diet, and 30% to other causes. (a) Which of the numbers mentioned above are conditional probabilities? 0.40 for diet 0.25 for cancer 0.30 for tobacco 0.30 for other causes (b)Find the probability that a death is due to cancer and tobacco.

Answer 1

Answer:

(a) P (T|C) = 0.30, P (T|D) = 0.40 and P (O|C) = 0.30

(b) The probability that a death is due to cancer and tobacco is 0.075.

Step-by-step explanation:

Let us denote the events as follows:

C = death is due to cancer

T = cause of cancer was tobacco

D = cause of cancer was diet

O = cause of cancer was other issue

The provided information is:

P (C) = 0.25

P (T|C) = 0.30

P (T|D) = 0.40

P (O|C) = 0.30

(a)

The conditional probability of an event A given that another event B has already occurred is denoted by, P (A|B).

This implies that the occurrence of event B is affected by the previously occurred event A.

Causes of cancer are tobacco, diet and other issues. If a death is caused by cancer then the probability that the cause of cancer was tobacco or diet or something else are conditional events.

The conditional probabilities are:

0.40 for diet, 0.30 for tobacco and 0.30 for other causes.

(b)

Conditional probability formula is:

$P (A|B)=\frac{P(A\cap B)}{P(B)}$

Compute the probability that a death is due to cancer and tobacco as follows:

$P (T|C)=\frac{P(C\cap T)}{P(C)}\\P(C\cap T)=P(T|C)\times P(C)\\=0.30\times0.25\\=0.075$

Thus, the probability that a death is due to cancer and tobacco is 0.075.