For a family with 2 children, the sample space indicating boy (B) or girl (G) is BB, BG, GB, and GG. If each of the outcomes is equally likely, find the probability that the family has 2 girls, given that the first child is a girl.
This is a problem of conditional probability. The shortcut to the answer is consider only the outcomes where the first child is a girl, and calculate the probability among those where both are girls.
I.E.
Ω={GG,GB}
favourable outcome: {GG}
So assuming all outcomes are equally probable, then
P(GG|first child is a girl)=1/2
The proper way is to use the definition:
P(GG|GX) ...both girls given first child is a girl
=P(GG∩GX)/P(GX)
=P(GG)/P(GX) .. since P(GG&GX)=P(GG) by absorption
=(1/4)/(1/2)
=1/2
Well, well, looks like we've got some probability puzzles going on here! So, let's break it down.
We know that the first child is a girl, which means we can exclude the outcome "BB" (as it requires that both children be boys). So now we're left with three possible outcomes: BG, GB, and GG.
Out of these three outcomes, there's only one outcome where the family has 2 girls, which is GG. Therefore, the probability of the family having 2 girls, given that the first child is a girl, is 1 out of 3.
But hey, let's not forget to have a little fun with this! So, the probability of the family having 2 girls in this scenario is like that one time when I tried to juggle chainsaws and accidentally caught them on fire – pretty unlikely, but not entirely impossible!
To find the probability that the family has 2 girls, given that the first child is a girl, we need to consider the favorable outcomes and the total outcomes.
The favorable outcomes are GG (2 girls) because we want both children to be girls.
The total outcomes are GB (girl, boy) and GG (2 girls) because the first child is a girl.
So, the probability that the family has 2 girls, given that the first child is a girl, is:
Number of favorable outcomes / Number of total outcomes
This is equal to:
1 (GG) / 2 (GB, GG)
Therefore, the probability is 1/2 or 0.5.
To find the probability that the family has 2 girls, given that the first child is a girl, we need to consider the possible outcomes where the first child is a girl.
Out of the four possible outcomes (BB, BG, GB, GG), the ones where the first child is a girl are BG, GB, and GG.
Now, we need to find the probability of having 2 girls among these three outcomes. The outcomes with 2 girls are GG.
So out of the three outcomes where the first child is a girl, only one outcome (GG) has 2 girls.
Therefore, the probability that the family has 2 girls, given that the first child is a girl, is 1 out of 3.