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Math Help

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1. Three cards are chosen at random from a standard deck of cards, without replacement. What is the probability that the first card is spades, the second card is hearts, and the third card is also spades?

2. Wu rolls two fair, six-sided dice. You are not told what the rolls were, but you are told that the sum of the two rolls is a prime. What is the probability that the sum of the two rolls is 5?

3. A bag contains five white balls and four black balls. Your goal is to draw two black balls.
a)You draw two balls at random. What is the probability that they are both black?

b)You draw two balls at random. Once you have drawn two balls, you put back any white balls, and redraw so that you again have two drawn balls. What is the probability that you now have two black balls? (Include the probability that you chose two black balls on the first draw.)

Mar 1, 2020

#1
+21955
+2

1)  spades x hearts x spades  =  13/52 x 13/51 x 12/50  =  3799/5100

2)  The primes are 2, 3, 5, 7, 11

2  occurs 1 way   (1,1)

3  occurs 2 ways  (1,2), (2,1)

5  occurs 4 ways  (1,4), (2,3), (3,2), (4,1)

7  occurs 6 ways  (1,6), (2,5), (3,4), (4,3), (5,2), (6.1)

11 occurs 2 ways  (5,6), (6,5)

There are 15 ways for a prime to be rolled of which 4 have a sum of 5:  probability  =  4/15

3)  a)  black x black  =  4/9 x 3/8  =  1/6

b)  You could get black x black  =  4/9 x 3/8  =  1/6

You could get black x white  =  4/9 x 5/8  =  5/18

Re-drawing the white and getting a black  =  3/8

Combining these two:  5/18 x 3/8  =  5/48

You could get white x black =  5/9 x 4/8  =  5/18

Re-drawing the white and getting a black  =  3/8

Combining these two:  5/18 x 3/8  =  5/48

You could get white x white  =  5/9 x 4/8  =  5/18

Re-drawing the two white and getting two black  =  4/9 x 3/8  =  1/6

Combining these two:  5/18 x 1/6  =  5/108

Adding these possibilities:  1/6 + 5/48 + 5/48 + 5/108  =  89/216

Mar 2, 2020
#2
+110167
+1

I have noted your lovely thank you to the forum. I really appreciate that.

Also could you name them appropriately,   like this one could be

"Probability,  -  cards and balls."

I know other people often don't do this but obviously I would like it if everyone did.

Mar 2, 2020