The Tuesday Birthday Problem

I ask people at random if they have two children and also if one is a boy born on a Tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.

Tip: Don't conclude too early. 

Click here to know the correct answer! 

Finding The Correct Probability

How tricky it was?

If you think that the probability is 1/2 after reading that the couple has equal chance of having child of either sex then you are in wrong direction.

Take a look at the table below.

There are 27 possible combinations when boy is born on Tuesday. Out of which there are only 13 possible combinations where either boy (first or second) is born on Tuesday. 

Hence the probability that the person having at least 1 boy off his 2 boys born on Tuesday is 13/27.

A Strange Liar

Richard is a strange liar. He lies on six days of the week, but on the seventh day he always tells the truth. He made the following statements on three successive days: 

Day 1: "I lie on Monday and Tuesday." 

Day 2: "Today, it's Thursday, Saturday, or Sunday." 

Day 3: "I lie on Wednesday and Friday."

On which day does Richard tell the truth? 

  Am I a liar?

Find the truth here! 

Source 

Truth Of a Strange Liar

What was his story? 

To find the truth we need to logical deduction here.

Now if statement on Day 1 is untrue then Richard must be telling the truth on Monday or Tuesday. 

And if Day 3 statement is untrue then he must be telling the truth on Wednesday or Friday. 

But he speaks true only on 1 day. So both statements of Day 1 & Day 3 can't be true at the same time. If so, then Richard speaks true on 2 days either Monday/Tuesday or Wednesday/Friday. This means that one of statements from Day 1 must be true & other must be untrue. That also makes the statement on Day 2 untrue always.

Case 1 : Day 3 statement is untrue.

In this case, Richard must be telling truth on either Wednesday or Friday. The statement on Day 1 would be true according to above logical deduction. Hence Day 2 must be either Thursday or Saturday. In both cases, statement on Day 2 would be true.

Case 2 : Day 1 statement is untrue.

If the statement made on Day 1 is untrue then Richard tells truth on Monday or Tuesday. Other statement on Day 3 must be true means Day 3 must be either Monday or Tuesday. If so, then Day 2 must be either Sunday or Monday. In case of Sunday, Day 2 statement would be true & in case of Monday Day 2 statement would be untrue. Hence Day 2 must be Monday & Day 3 must be Tuesday.

So Richard tells truth on Tuesday.