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# Pilgrim's Probability Conundrums watch

1. Should be a walk in the park for maths students. See how you do.

Puzzle 1

Mary has 2 children. One of them is a boy.
What is the probability of her other child also being a boy?

Puzzle 2

A married couple have 4 children.
There could be 4 children of the same sex (all girls, or all boys)
There could be 3 of one sex and one of the other
Or there could be 2 of each sex.

Which is the most likely ?
2. 1. 0.25
2. 2 of each sex

Hopefully that's right and my brain hasn't died yet!
3. Sorry to say no, wrong on both puzzles :-(
4. (Original post by PilgrimOfTruth)
Sorry to say no, wrong on both puzzles :-(
Haha oops, maybe I'll try again when I'm not so tired.
5. i) P (BB|B) = (1/4) / (3/4) = 1/3
6. ii) bbbb or gggg = 1/16 + 1/16 = 2/16

bggg or gbbb ( any order ) = 4/16 + 4/16 = 8/16

bbgg ( any order ) = 6/16

so 3 and 1 is the most likely.
7. 1) The probabilities are independent, so 50% (or 51% or whatever the actual birth rate is)
2) 3 of one and one of the other, simple combinatorics
8. The Bear has it all correct. Congrats and Kudos.

Answer 1 - is 1 in 3

The possible scenarios for 2 children are

BB, BG, GB, GG

Since we know one is a boy then option "GG" is excluded. That leave 3 options of which only one is BB so the answer is 1 in 3 chance.

Answer 2 - is the 3 of one sex and 1 of the other is most likely

The possible combinations of different sexes are 16 as follows:

BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG,
GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG

Of these 2 are "all same sex", 6 are "2 of each sex" and 8 are "3 of one and 1 of the other sex" so the latter is the most likely.
9. Puzzle 3

What 2 numbers come next in this logical sequence ?

3, 3, 5, 4, 4

Puzzle 4

A man shows you a cloth bag containing a marble, and explains that the marble inside the bag is either black or white (but you can't see which).* He then adds a white marble, shakes the bag, and takes out a marble at random. It’s white.* *

The question is this. What are the odds that the remaining marble is white?*
10. (Original post by PilgrimOfTruth)
The Bear has it all correct. Congrats and Kudos.

Answer 1 - is 1 in 3

The possible scenarios for 2 children are

BB, BG, GB, GG

Since we know one is a boy then option "GG" is excluded. That leave 3 options of which only one is BB so the answer is 1 in 3 chance.
This isn't accurate the way you've expressed it, which prescribes that one or the other is male. If you're enumerating the options (which seems an overly lengthy way to solve), taking 'B' to be the one which is prescribed as male, 'b' to be the other boy, and 'g' to be the other girl, the options are:

Bb, bB, Bg, gB.

It's simpler just to take the approach of discarding one from consideration at random, which leaves two options - b and g.
11. (Original post by PilgrimOfTruth)
Puzzle 3

What 2 numbers come next in this logical sequence ?

3, 3, 5, 4, 4
I would guess at 6, 5; but really there isn't enough information given for there to be a single correct answer.

Puzzle 4

A man shows you a cloth bag containing a marble, and explains that the marble inside the bag is either black or white (but you can't see which).* He then adds a white marble, shakes the bag, and takes out a marble at random. It’s white.* *

The question is this. What are the odds that the remaining marble is white?*[/QUOTE]

Let's start with the initial composition of the marbles after the white marble is added. P(w,b) = 50%, P(w,w) = 50%.

A white marble being removed from the former is half as likely as the latter. Therefore, given a white marble was removed, P(w,w) = 2/3.
12. @TheDefiniteArticle

Puzzle #3 - incorrect

Puzzle #4 - Correct

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Updated: January 21, 2016
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