Quarter Wit, Quarter Wisdom: Probability with Conditions Part II

Quarter Wit, Quarter WisdomLast week I left you with a conditional probability question. Let’s look at its solution now. This will be my last post on GMAT Combinatorics and Probability (for a while at least) until and unless you want me to take up a particular concept/question related to this topic. Next week, we will start a new topic.

Back to question at hand:

Question 2: Alex has five children. He has at least two girls (you do not know which two of her five children are girls). What is the probability that he has at least two boys too? (The probability of having a boy is 0.4 while the probability of having a girl is 0.6)

Solution:

We want to find this probability: P(‘At least 2 Boys and at least 2 Girls’ given ‘At least 2 Girls’) = P(At least 2 Boys and at least 2 Girls)/P(At least 2 Girls)

Let’s try and find P(At least 2 Boys and at least 2 Girls) and P(At least 2 Girls)

‘At least 2 Boys and at least 2 Girls’ can be obtained in two ways: ‘3 Boys and 2 Girls’ or ‘2 Boys and 3 Girls’

P(At least 2 Boys and at least 2 Girls) = P(3 Boys and 2 Girls) + P(2 Boys and 3 Girls)

P(2 Boys and 3 Girls) = 0.4*0.4*0.6*0.6*0.6 * 5!/(2!*3!) = (0.4)^2 * (0.6)^3 * 10

You multiply by 5!/(2!*3!) because out of the five children, any 2 could be boys and the other three would be girls. So you have to account for all arrangements: BBGGG, BGBGG, GGBGB etc

P(3 Boys and 2 Girls) = 0.4*0.4*0.4*0.6*0.6 * 5!/(3!*2!) = (0.4)^3 * (0.6)^2 * 10

P(At least 2 Boys and at least 2 Girls) = [(0.4)^2 * (0.6)^3 * 10] + [(0.4)^3 * (0.6)^2 * 10] = (0.4)^2 * (0.6)^2 *10 (0.6 + 0.4) = (1.6)(0.36)

Now that we have P(At least 2 Boys and at least 2 Girls), let’s focus on getting P(At least 2 Girls). Again, as we saw last week, there are 2 ways of arriving at P(At least 2 Girls).

P(At least 2 Girls) = P(2 Girls and 3 Boys) + P(3 Girls and 2 Boys) + P(4 Girls + 1 Boy) + P(5 Girls)

OR

P(At least 2 Girls) = 1 – P(5 Boys) – P(1 Girl and 4 Boys)

Let me show you the calculations involved in both the methods.

Method 1:

P(At least 2 Girls) = P(2 Girls and 3 Boys) + P(3 Girls and 2 Boys) + P(4 Girls + 1 Boy) + P(5 Girls)

P(2 Girls and 3 Boys) = (0.4)^3 * (0.6)^2 * 10 (from above)

P(3 Girls and 2 Boys) = (0.4)^2 * (0.6)^3 * 10 (from above)

P(4 Girls + 1 Boy) = (0.4)*(0.6) *(0.6)*(0.6)*(0.6)*5!/4! = (0.4) * (0.6)^4 * 5

P(5 Girls) = (0.6)*(0.6)*(0.6)*(0.6)*(0.6) = (0.6)^5

P(At least 2 Girls) = [(0.4)^3 * (0.6)^2 * 10] + [(0.4)^2 * (0.6)^3 * 10] + [(0.4) * (0.6)^4 * 5] + [(0.6)^5]

Method 2:

P(At least 2 Girls) = 1 – P(5 Boys) – P(1 Girl and 4 Boys)

P(5 Boys) = (0.4)* (0.4)* (0.4)* (0.4)* (0.4) = (0.4)^5

P(1 Girl and 4 Boys) = (0.6)* (0.4)*(0.4)*(0.4)*(0.4)*5!/4! = (0.6)*(0.4)^4 * 5

P(At least 2 Girls) = 1 – [(0.4)^5] – [(0.6)*(0.4)^4 * 5]

The values in bold are the same even if they don’t look same. (Trust me, I checked on my financial calculator!)

P(‘At least 2 Boys and at least 2 Girls’ given ‘At least 2 Girls’) = P(At least 2 Boys and at least 2 Girls)/P(At least 2 Girls)

P(‘At least 2 Boys and at least 2 Girls’ given ‘At least 2 Girls’) = (1.6)(0.36)/[1 – (0.4)^5 - (0.6)*(0.4)^4 * 5]

Even though the solution looks complicated, I hope you see that the approach is quite logical and straight forward. Let’s bid farewell to combinatorics and probability now. We will take up some other topic next week. Till then, keep practicing!

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

4 Responses

  1. sb says:

    If it is already known that there are at least 2 girls, why do we put the probabilities of these 2 girls in the calculation?

    • Karishma says:

      When you say there are at least 2 girls, what all combinations are possible?
      There are exactly 2 girls
      There are exactly 3 girls
      There are exactly 4 girls
      There are exactly 5 girls

      You need to consider the case where there are exactly 2 girls and the cases where there are more than 2 girls.

  2. sb says:

    I should have provided more description on what I was asking. Your solution is correct and we need to consider the 4 cases you listed in your reply.

    I am not able to understand conceptually why you are accounting for the probability of the 2 girls, who we know are already present, in each step of your calculation. As an example, let’s take this step:
    P(3 Boys and 2 Girls) = 0.4*0.4*0.4*0.6*0.6 * 5!/(3!*2!) = (0.4)^3 * (0.6)^2 * 10

    Do I need to multiply 0.6 two times in the above step and the other steps? We already know that there are 2 girls.

    • Karishma says:

      I think you are unsettled by the concept of conditional probability here. You are already given that there are 2 girls so why are we accounting for them? Is that it?

      Notice the first line of the solution:
      We want to find this probability: P(‘At least 2 Boys and at least 2 Girls’ GIVEN ‘At least 2 Girls’)

      How do we find this?
      P(A given B) = P(A)/P(B)
      This is discussed in part 1 of this post: http://www.veritasprep.com/blog/2012/02/quarter-wit-quarter-wisdom-probability-with-conditions/

      Now to calculate P(A given B), you need to find P(B) i.e. we need to find P(atleast 2 girls)

      It’s like, we know that there are 2 girls but what was the probability of that happening? Now that that has happened, out of those cases, what is the probability that there will be 2 boys too. Hence, calculating P(atleast 2 girls) is a step in calculating the desired probability.

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