Ball Paradox

Probability Level 2

A box initially contains a ball, which has a 50:50 chance of being black or white.

Then you put a white ball into the box, and randomly pick one ball out, which turns out to be a white ball.

What is the probability that the initial ball was white?

Note: This problem was stated by the famous author, Lewis Carroll (Charles Dodgson).

\dfrac{1}{4}

\dfrac{1}{3}

\dfrac{1}{2}

\dfrac{2}{3}

\dfrac{3}{4}

1

2 solutions

Geoff Pilling
Nov 17, 2016

I used Baye's Theorem:

$P(W_1|W_2) = \frac{P(W_2|W_1)P(W_1)}{P(W_2)} = \frac{1 \cdot \frac{1}{2}}{\frac{3}{4}} = \boxed{\frac{2}{3}}$

Where:

$W_1$ = The event that the ball initially in the box was white.
$W_2$ = The event that the ball picked was white.

I dont understand why P(W2) = 3/4. Pls explain.

Alice Cristabel - 3 years, 10 months ago

Worranat Pakornrat
Nov 16, 2016

Since there's 50:50 chance that the original ball is black or white, there are two possible scenarios after putting the white ball in (B for black and W for white):

{B , W} or {W , W}

For {B , W} case, there are two possible drawing outcomes: first draw is black then white; or first is white then black. However, since the first draw is white, the former outcome is invalid, making the draw to be first white then black.

For {W , W} case, the two possible drawing outcomes will both be first and second draw are white.

Totally there are $3$ possible outcomes with first white drawing, and $2$ of these outcomes have white as original color. (One outcome has black as in the first case.)

Therefore, the probability of white ball originally = $\dfrac{2}{3}$ .

Ball Paradox

2 solutions

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