1.Determine the poker odds of drawing the following hand from a standard 52 card

Question

1.Determine the poker odds of drawing the following hand from a standard 52 card deck. (4 suits, 13 ranks in each suit.)

What are the odds of drawing 4 of a kind? (Drawing 4 cards of the same rank plus 1 other non-matching card.) Draw the cards in the following order XXXXY. Then compute how many different ways the 5 cards can be ordered. Drawing cards is also without replacement. (NOTE: If necessary, lay out 52 cards on a table and do a dry run before computing the probabilities!)

a.What is the probability of drawing one of the possible matching cards X for your first card? (Remember, at this point, X can be any rank.)

b.What is the probability of drawing a 2nd card of the same rank X that matches the rank of your 1st card?

c.What is the probability of drawing a matching card X for your 3rd card?

d.What is the probability of drawing another matching card X for your 4th card?

e.What is the probability of drawing a non-matching card Y for your 5th card? (Y does not match the 4 X’s.)

f.Now we drew the cards in a specific order (XXXXY). But the order of the drawing of the cards does not matter in the end.

We could have drawn the cards as XXXYX, XXYXX, etc. as still ended up with 4 of a kind. If we use the reordering formula, then we would get

So what number do you need to multiply by to get the final probability which takes into account that drawing order does not matter?

            g.   Multiply the above numbers (a) to (f) to get the final probability. Express you answer as 1/p (i.e., .10 = 10, .5 = 2)

(Note: The probability of getting 1 pair are 1/p = 4165. If you did not get close to this number, then you did something wrong.)

Explanation / Answer

a) 1       {choose any suits}

b) 3/51   { there are 3 card of same suit left out of 51 remaining card}

c)2/50 { there are 2 card of same suit left out of 50 remaining card}

d)1/49 { there are 1 card of same suit left out of 49 remaining card}

e)1        {select any card out of 48 remaining

f)5            {5!/(4!)= 4

g)3/51*2/50*1/49*5

=1/4165

odds = p/(1-p)=1/4164

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