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# Category Archives: Drawing Balls

## Boxes with Yellow and Green Balls

Three boxes contain yellow and green balls

- Box 1 contains 2 yellow balls.
- Box 2 contains 2 green balls.
- Box 3 contains 1 yellow ball and 1 green ball.

One box is selected at random, and one ball is pulled out of that box.

- The ball that is pulled out of the chosen box is yellow. What is the probability that the other ball in that same box is also yellow?
- Let \(A\) be the event that Box 3 is chosen. Let \(B\) be the event that a yellow ball is pulled out of the chosen box. Are \(A\) and \(B\) independent?
- The ball that is pulled out of the chosen box is yellow. Without replacement, a second ball is chosen at random from one of the three boxes. (Each box has a 1/3 chance of being selected.) What is the probability that the second ball chosen is also yellow?

## Ball in Boxes

Suppose you have three boxes, \(Box_1,Box_2,Box_3\), such that \(Box_i\) contains \(i\) white balls and one black ball.

You will to select one ball from the boxes. Here are two schemes you could use for selection:

- Select one box uniformly at random. Pull one ball from that box. Or,
- Dump all the balls into one box. Mix them up. Pull out one ball.

Are these two schemes probabilistically equivalent?

Suppose instead of selecting a box uniformly at random, you select \(Box_i\) with probability \(p_i\). Find a list of values for \(p_1, p_2,\) and \(p_3\) that would make this new scheme probabilistically equivalent to scheme 2?

## Picking a box then a ball

Suppose that there are two boxes, labeled odd and even. The odd box contains three balls numbered 1,3,5 and the even box contains two balls labeled 2,4. One of the boxes is picked randomly by tossing a fair coin.

- What is the probability that a 3 is chosen ?
- What is the probability a number less than or equal to 2 is chosen ?
- The above procedure produces a distribution on \(\{1,2,3,4,5\}\) how does it compare to picking a number uniformly (with equal probability) ?

[Pitman p 37, example 5]

## Balls in a Box: Counting

A box contains 20 red balls and 30 black balls. Four balls are chosen without replacement. What is the chance that:

- all balls are red
- exactly three balls are red
- the first red ball appears on the last draw.
- the fist two balls are the same color

## Drawing tickets

A box contains tickets marked \(1,2,…,n\). A ticket is drawn at random from the box.

Sampling with replacement — Then the ticket is replaced in the box and a second ticket is drawn at random. Find the probability of the following events:

a) the first ticket drawn is numer 1 and the second is number 2;

b) the numbers on the two tickets are consectutive integers;

c) the second number drawn is bigger than the first number.

Sampling without replacement — The ticket is not replaced in the box and a second ticket is drawn at random.

d) Repeat a)-c).

[Pitman page 9, Problem 3]

## Balls in boxes

A box contains 1000 bulbs, of which 2 are black and the rest are white.

- Which of the following is mostlikely to occur in 1000 draws with replacement from the box ? Fewer than 2 black balls, Exactly two black balls, more than two black balls.
- If two series of 1000 draws are made at random from the box, what is, approximately, is the chance that they produce the same number of black balls ?