Probability Question?

Question

In a vase, there are 2 red balls, 4 blue balls and a red ball. You have to pick 3 balls without replacement. How many blue balls do you expect to remove?

P.S.:- please provide method too, not just the answer...

Answer 1

LithuimD is exactly correct, but he didn't finish the problem. He told you how to compute the probabilities of each of the 4 outcomes, but didn't tell you how to compute the expected number removed. If we let k = the number of blue balls picked out of 3, we see k can equal 0, 1, 2, or 3 (which is 4 possible outcomes). Expected number, E(k) is defined as:

E(k) = Sum(k*P(k)), for all k.

Before we use LithuimD's equation, I must note that there seems to be a mistake in your statement of the problem. You say there are 2 red balls and then say "... and a red ball." I assume you meant a color other than red or blue.

Using that assumption, and then using LithuimD's equation, we find that P(0) = 1/35, P(1) = 12/35, P(2) = 18/35, and P(3) = 4/35). So,

E(k) = 0*(1/35) + 1*(12/35) + 2*(18/35) + 3*(4/35) = 12/7

So we expect, on the average to have 1.714 blue balls.

Answer 2

X = no of blue balls
RTF E(X)

P(X = 3) = 4/7 * 3/6 * 2/5 = 4/35
P(X = 2) = 3 * 4/7 * 3/6 * 3/5 = 18/35
P(X = 1) = 3 * 4/7 * 3/6 * 2/5 = 12/35
P(X = 0) = 3/7 * 2/6 * 1/5 = 1/35

E(X) = ∑ x*P(X = x)
= 3*4/35 + 2*18/35 + 1*12/35 + 0
= 60/35 = 12/7

You expect to get 12/7 blue balls
Note that the expected value does not have to be a whole number.

Answer 3

You have 4 blue balls out of 7 balls in total, and you're picking three of them without replacement.

Whenever you see the phrase "without replacement", such as in pulling items out of a jar or drawing cards from a deck, the phrase "hypergeometric distribution" should leap to mind (at least, in the future it should).

The Microsoft Excel function is

=hypgeomdist(A,B,C,D)

where
A = the number of items you're picking
B = the number of copies of the desired item
C = the total number of items
D = the exact number of copies of the desired item you want to obtain

The function will tell you the probability of getting exactly D of the desired item when you're picking A items at random without replacement, when there are B copies of the desired item out of C total items. Simply put that into Excel (without the quotation marks) and replace the variables with the appropriate integers.

The calculation it's actually performing is this:

com[B, D] * com[(C - B), (A - D)] / com[C, A]

where com[] indicates the combination function:

com[x, y] = x! / (y! * (x - y)!)

So, plugging in your numbers, we end up with
hypgeomdist(A,B,C,D)
hypgeomdist(3,4,7,D)
We do this for D = 0 to find the probability of getting exactly 0 blue balls, for D = 1 to find the probability of getting exactly 1 blue ball, etc.

After performing the calculations, we find that the probabilities of getting various numbers of blue balls are:
0 blue balls = 0.028571429
1 blue ball = 0.342857143
2 blue balls = 0.514285714
3 blue balls = 0.114285714

It becomes clear that the number you'd most likely get (51.4% of the time) is 2 blue balls when picking 3 balls out of 7 balls, out of which 4 are blue.