Question

1). If a couple has two children, what is the probability that they are both girls assuming that the older one is a girl?

2). Suppose that we have two dice, the first one being a regular die, and the second weighted so that half the time it rolls a 1, and half the time it rolls a 2 (it never rolls anything else). If we choose one of these dice at random, and roll a 1, what’s the probability that it is the regular die?

Answer 1

To find the probability that both are females, if we know that the elder child is female. Sample Space= S = {MM, MF, FM, FF},

Let A: event that both are females A: {FF}

Let B: event that elder one is a female child.B: {FM, FF}

Hence→ P (A ∩ B) = {FF}

P (both are females given the elder child is female) =P(A/B)=(P(A∩B)/P(B))

Given S = {MM, MF, FM, FF}, we can see that: P (A) = 1/4; P (B) = 2/4=1/2; P (A ∩ B) = 1/4

Therefore, P(A/B)=(P(A∩B)/P(B))=2/4=1/2

2) Since 1st die is a regular one, probability of getting number 1 after the roll= 1/6

Since 2nd die is not a regular one, probability of getting number 1 after the roll= 1/2

Probability of Choosing a regular Die P(R)= 1/2

Probability of getting 1 on die= P(1)=

PROBABILITY OF 1 IN REGULAR DICE P(1/R)= 1/6

PROBABILITY OF DICE ROLLED BEING REGULAR P(R)=1/2

TOTAL PROBABILITY OF ROLLING 1= P(1)=1/6+1/2=4/6=2/3

P (the dice rolled is regular one given no. rolled is 1) =

=1/8