You play the following game against your tiend. You have 2 urns and 5 balls. One

Question

You play the following game against your tiend. You have 2 urns and 5 balls. One of the balls is black and the other 4 are white. You can place the balls in the ums any way that you'd ike, including leaving an urn empty. Your friend will choose one un at random and then draw a ball from that urn (f he nothing) She wins if she draws the black ball and loses otherwise. chooses an empty un, he draws a Suppose you arrange the balls in the way that minimizes her chances of drawing the black bal. What is her probability of winning? Probability you arrange the balls in the way that maximizes her chances of drawing the black ball What is her chance of winning? Probability- c. What are the analogous probabilities when there are n balls total (one black, n 1 white) instead of 5 balls total? Minimizing probability- Maximizing probability

Explanation / Answer

(a) we will keep all the 5 balls in a single urn and leave the other urn empty.

probability of choosing urn = 1/2, as their are 2 urns

Probability of choosing black ball from the empty earn = 0

Probability of choosing black ball from the filled urn = 1/5

Now, probability of winning = 1/2 × 0 + 1/2 × 1/5 = 1/10

Answer :- 1/10

(b) we will keep 1 ball in one urn and the other 4 balls in other urn.

Probability of choosing urn = 1/2

Probability of choosing ball from the urn which has 1 ball = 1

Probability of choosing ball from the urn which has 4 balls = 1/4

Probability of winning = 1/2 × 1 + 1/2 × 1/4 = 1/2 + 1/8 = 5/8

Answer :- 5/8

(c) similarly,

For maximising probability, as above, we will split the balls as 1 in one urns and the rest other balls in other urn

Probability of choosing ball from urn with one ball = 1

Probability of choosing ball from urn with n-1 balls = 1/n-1

Probability of winning = 1/2 × 1 + 1/2 × 1/n-1 = n/2(n-1)

And,

For minimising probability, we will keep all the balls in a single urn and leave the other urn empty

Probability of choosing ball from the urn having 0 ball = 0

Probability of choosing ball from the urn with n balls = 1/n

Probability of winning = 1/2 × 0 + 1/2 × 1/n = 1/2n

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