An urn contains 4 red balls numbered from 1 to 4 and 6 white balls numbered from 5 to 10.we choose simultaneously 2 balls fron the urn.?

calculate the probability of the following events:

-the drawn balls are red

-the drawn balls have different colors

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## Answers

PART 1 - The two drawn balls are red

On the first draw, there are 4 balls out of 10 that are red:

P(first is red) = 4/10 = 2/5

On the second draw, there are 3 balls out of 9 that are red:

P(second is red) = 3/9 = 1/3

Multiply to get the probability of both events:

P(both are red) = 2/5 x 1/3 = 2/15

Answer:

2/15

PART 2 - The two drawn balls are different colors.

Let's first figure out the probability of drawing a red ball and then a white ball:

On the first draw, there are 4 balls out of 10 that are red:

P(first is red) = 4/10 = 2/5

On the second draw, there are 6 balls out of 9 that are white:

P(second is white) = 6/9 = 2/3

Multiply to get the probability of both events:

P(first is red, second is white) = 2/5 x 2/3 = 4/15

But we have a second, equivalent case where the first ball could have been white and the second ball could have been red. It has the same probability of 4/15.

If you aren't convinced we can figure the separate probability as:

On the first draw, there are 6 balls out of 10 that are white:

P(first is white) = 6/10 = 3/5

On the second draw, there are 4 balls out of 9 that are red:

P(second is red) = 4/9

Multiply to get the probability of both events:

P(first is white, second is red) = 3/5 x 4/9 = 12/45 = 4/15

Add both cases:

4/15 + 4/15

= 8/15

Answer:

8/15

P.S. If you are familiar with the "n choose k" formula, you can do this even quicker.

PART 1:

From the 4 red, choose 2 --> 4C2 = (4 x 3) / (2 x 1) = 6 ways

From the 10 balls, choose *any* 2 --> 10C2 = (10 x 9) / (2 x 1) = 45 ways

Divide to get the probability:

6/45 = 2/15

PART 2:

From the 4 red, choose 1 --> 4C1 = 4 ways

From the 6 white, choose 1 --> 6C1 = 6 ways

Total ways to pick 1 red and 1 white = 4 x 6 = 24 ways

From the 10 balls, choose *any* 2 --> 10C2 = 45 ways

Divide to get the probability:

24/45 = 8/15

draw R p = 4/10

then draw R p = 3/9

p = (4/10)(3/9) = 2/15

answer is 2/15

draw R p= 4/10

then draw W = 6/9

p = (4/10)(6/9) = 4/15

Or

draw W p= 6/10

then draw R p = 4/9

p = (6/10)(4/9) = 4/15

p (different colors) = 4/15+4/15 = 8/15

draw W p = 6/10

then draw W p = 5/9

p = (6/10)(5/9) = 5/15 = 1/3

RR = 2/15

RW = 4/15

WR = 4/15

WW = 5/15