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# Tricky Probability question

Wed 12 Apr, 2017 12:00 pm
There are 100 balls: 80 Black and 20 White.
At random, 5 balls are placed in a bag.
1 ball is then taken out.
What is the probability the ball is white?
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Type: Question • Score: 1 • Views: 1,125 • Replies: 7
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fouchard

-1
Fri 14 Apr, 2017 03:26 pm
@shunt,
p=1/100
0 Replies

markr

2
Sun 16 Apr, 2017 02:54 am
@shunt,
1/5
roger

1
Sun 16 Apr, 2017 12:59 pm
@markr,
Especially if four are black and one is white, but what's the probability of that?
markr

2
Sun 16 Apr, 2017 02:51 pm
@roger,
The intermediate step is irrelevant. By selecting one of 100, we're "unselecting" 99. So, let's do 99 unselections, one at a time, instead of one selection. However, after we've done 95, let's throw the remaining five balls into a bag and follow that up with four more unselections. The process is equivalent to selecting one of 100, so the probability of the selected ball being white is 20/100.
roger

1
Sun 16 Apr, 2017 02:57 pm
@markr,
You don't have a hundred. You have five. For all we are given, you may have five black balls in you bag. Probably not, but possible.
markr

1
Sun 16 Apr, 2017 04:22 pm
@roger,
You start with 100, then place five in a bag. The percentage of white balls in the initial set of 100 is the probability of drawing a white ball from the bag when calculated prior to taking any steps. Obviously, if you get to see what goes into the bag prior to calculating the probability, it may be change.
0 Replies

Kolyo

1
Sun 16 Apr, 2017 05:26 pm
The probabilities that there will be a certain number of white balls in the bag after the first step are:

Prob. of '0': 0.319309
Prob. of '1': 0.420144
Prob. of '2': 0.207344
Prob. of '3': 0.047849
Prob. of '4': 0.005148
Prob. of '5': 0.000206

(source: http://stattrek.com/online-calculator/hypergeometric.aspx)

The probabilities that there will be a certain number of white balls in the bag after the first step, and then we will subsequently draw a white ball on the second step are:

Probability 0 are in the bag, and we draw a white one: 0.319309*(0/5) = 0
Probability 1 is in the bag, and we draw a white one: 0.420144*(1/5) = 0.084029
Probability 2 are in the bag, and we draw a white one: 0.207344*(2/5) = 0.082938
Probability 3 are in the bag, and we draw a white one: 0.047849*(3/5) = 0.028709
Probability 4 are in the bag, and we draw a white one: 0.005148*(4/5) = 0.004118
Probability 5 are in the bag, and we draw a white one: 0.000206*(5/5) = 0.000206

The probability that we will ultimately draw a white ball is this second set of figures, all added together, which is 0.2.

So markr's sleight-of-hand approach gets him to the right answer in the end.
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