Question 1206285: An urn contains 5 red balls and 8 yellow balls. If Vince chooses 10 balls at random from the urn, what is the probability that he will select 4 red balls and 6 yellow balls? Round your answer to 3 decimal places.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the set contains 5 red and 8 yellow balls.
i think this can be done this way.
p(4 red and 6 yellow) = c(5,4) * c(8,6) / c(13,10) = .4895104895.
i think it can also be done this way.
p(4 red and 6 yellow) = 5/13 * 4/12 * 3/11 * 2/10 * 8/9 * 7/8 * 6/7 * 5/6 * 4/5 * 3/4 * c(10,4) = .4895104895.
the first way is essentially the number of ways you can get your want divided by the total number of possible ways.
the formula is p(4 red and 6 yellow) = c(5,4) * c(8,6) / c(13,10)
that translates to:
the number of ways you can get 4 red balls from 5 red balls is c(5,4).
the number of ways you can get 6 yellow balls from 8 yellow balls is c(8,6).
the number of ways you can get 4 red balls and 6 yellow balls becomes c(5,4) * c(8,6) which translates to the number of ways you can get 4 red balls times the number of ways you can get 6 yellow balls.
that's the numerator.
the number of possible ways to get 10 balls out of 13 balls is c(13,10).
that's the denominator.
the second way looks at the probability of getting them in a specified order.
that formula becomes p(4 red and 6 yellow) = 5/13 * 4/12 * 3/11 * 2/10 * 8/9 * 7/8 * 6/7 * 5/6 * 4/5 * 3/4 * c(10,4) = .4895104895.
on your first draw you pick a red.
the probability of that is 5/13.
on your second draw you pick another red.
the probability of that is 4/12.
ditto until you pick the 4th ball as red.
starting from the 5th ball you draw, you are not looking at yellow.
probability of the firth ball being yellow is 8/9.
that's because there are 8 yellow balls left out of a total of 9 left.
this continues until your last draw of a yellow ball whose probability is 3/4.
there are 4 balls left.
they are RYYY
that's because you took our 4 red and 5 yellow for a total of 9
13 minus 9 = 4 balls left
5 red minus 4 red = 1 red left
8 yellow minus 5 yellow = 3 yellow left.
you wind up with 10 balls that were picked in a certain order.
the number of ways those 10 could be ordered is c(10,4).
that's the number of possible ways you can arrange 4 red balls from 10 total balls.
the other number of ways those 10 could be ordered is c(10,6).
that's the number of possible ways you can arrange 6 yellow balls from 10 total balls.
c(10,4) = 210
c(10,6) also = 210.
two different ways get you the same answer which at least confirms the ways are consistent.
to see how each formula works, you can work a different problems where the numbers are small enough that you can show each possible say separately.
i did that with this problem and it confirms both methods are good.
your solution is that the probability is equal to .4895104895.
round that to 3 decimal places to get .490 which is the same as .49.
even though .49 is the same as .490, you should probably show it as .490 just to indicate that you rounded to 3 decimal places, and not 2.
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