Thread: Probability and Stats Question!!!

John, who is 7 years old, is mapping out his future. He thinks that with probability

2/5 he might
become an astronaut (event A), with probability 1/4 he might be a basketball star (event B), and with probability 1/2 he might be happily married and have lots of children (event C). He accepts that he
can’t be an astronaut and a basketball star at the same time but thinks that astronauts and basketball
stars are just as likely as anyone else to have lots of children.
(i) Draw a Venn diagram illustrating this situation.
(ii) Calculate the probability that John has lots of children but does not get to be an astronaut
or a basketball star.
(iii) What is the probability that none of John’s dreams comes true?
[6 marks]

4. A test for a disease has probability 0.96 of giving a positive result when the disease is in fact present,
and probability 0.99 of giving a negative result when the disease is absent.
(i) About 1 in 200 adults have the disease. If a randomly selected adult is tested for the disease,
what is the probability of a positive result?
(ii) A doctor believes there is a 25% chance that his patient has the disease. The patient is tested
and the test comes up positive. What is now the probability that the patient has the disease?

[6 marks]

Help On either of these questions would be great!!!
Thanks

Originally Posted by agrabham

John, who is 7 years old, is mapping out his future. He thinks that with probability

2/5 he might

become an astronaut (event A), with probability 1/4 he might be a basketball star (event B), and with probability 1/2 he might be happily married and have lots of children (event C). He accepts that he
can’t be an astronaut and a basketball star at the same time but thinks that astronauts and basketball
stars are just as likely as anyone else to have lots of children.
(i) Draw a Venn diagram illustrating this situation.
(ii) Calculate the probability that John has lots of children but does not get to be an astronaut
or a basketball star.
(iii) What is the probability that none of John’s dreams comes true?
[6 marks]

[snip]

There's not enough information for a unique answer IF the to (iii) is assumed to be non-zero. To draw the Venn diagram, draw a circle labelled C and let it intersect with circles labelled A and B. The circles labelled A and B do not intersect. Let Pr(A and C') = a. It looks like you can assume that Pr( A and C) = Pr(A and B) = b, say. Let Pr(C and A' and B') = c. Let Pr(B and C') = d. Assuming the answer to (iii) is zero then:

a + b = 2/5 .... (1)
b + d = 1/4 .... (2)
2b + c = 1/2 .... (3)
a + 2b + c + d = 1 .... (4)

I get a = 0.325, b = 0.075, c = 0.35, d = 0.175.

Originally Posted by agrabham

[snip]

4. A test for a disease has probability 0.96 of giving a positive result when the disease is in fact present,
and probability 0.99 of giving a negative result when the disease is absent.
(i) About 1 in 200 adults have the disease. If a randomly selected adult is tested for the disease,
what is the probability of a positive result?
(ii) A doctor believes there is a 25% chance that his patient has the disease. The patient is tested
and the test comes up positive. What is now the probability that the patient has the disease?

[6 marks]

Help On either of these questions would be great!!!
Thanks

Draw a tree diagram. Start with branches for D and D'. Then for each, have a branch for +ve and -ve. Note that Pr(D) = 1/200 = 0.005. Therefore Pr(D') = 1 - 0.005 = 0.995.

(i) From the tree diagram I get Pr(+ve) = (0.005)(0.96) + (0.995)(0.01) = ......


(ii) Edit the tree diagram by:

Replace Pr(D) = 0.005 with Pr(D) = 0.25.
Replace Pr(D') = 0.995 with Pr(D') = 0.75.

Pr(D | +ve) = Pr(D and +ve)/Pr(+ve).

From the tree diagram I get Pr(D | +ve) = (0.25)(0.96)/[(0.25)(0.96) + (0.75)(0.01)] = ....