Question

2. In a game, a player throws a four-sided fair die repeatedly until he rolls a 1, 2, or 3. In other words, the player continues to throw the die as long as he rolls 4s. When he rolls a "non-4," he stops. a. What is the probability that the player tosses the die exactly three times? b. What is the expected number of rolls needed to obtain the first non-4? c. If he rolls a non-4 on the first throw, the player is paid $5. The player is paid $10 if he rolls a 4 followed by a non-4; $20 if he rolls two 4s followed by a non-4; $30 if he rolls three 4s followed by a non-4; and $50 in all other cases. What is the expected amount paid to the player?

          2. In a game, a player throws a four-sided fair die repeatedly until he rolls a 1, 2, or 3. In other words, the player continues to throw the die as long as he rolls 4s. When he rolls a "non-4," he stops.
a. What is the probability that the player tosses the die exactly three times?
b. What is the expected number of rolls needed to obtain the first non-4?
c. If he rolls a non-4 on the first throw, the player is paid $5. The player is paid $10 if he rolls a 4 followed by a non-4; $20 if he rolls two 4s followed by a non-4; $30 if he rolls three 4s followed by a non-4; and $50 in all other cases. What is the expected amount paid to the player?

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Added by Jose Miguel B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Sri K

03/17/2023

Step 1

To roll the die exactly three times, the player must roll a 4 on the first two rolls and then roll a non-4 on the third roll. The probability of rolling a 4 is $\frac{1}{4}$, and the probability of rolling a non-4 is $\frac{3}{4}$. So the probability of this Show more…

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2. In a game, a player throws a four-sided fair die repeatedly until he rolls a 1, 2, or 3. In other words, the player continues to throw the die as long as he rolls 4s. When he rolls a "non-4," he stops. a. What is the probability that the player tosses the die exactly three times? b. What is the expected number of rolls needed to obtain the first non-4? c. If he rolls a non-4 on the first throw, the player is paid $5. The player is paid $10 if he rolls a 4 followed by a non-4; $20 if he rolls two 4s followed by a non-4; $30 if he rolls three 4s followed by a non-4; and $50 in all other cases. What is the expected amount paid to the player?

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Transcript

00:01 Hello, let's have a look on the question.

00:02 So it is given to us that we have a four -sided dice having 1, 2, 3 and 4 written on it.

00:08 So from here we can say that probability of 1, 2, 3 and 4 coming up will be equal to 1 upon 4.

00:14 So for the a part, we need to find out that he will stop after exactly 3 throats.

00:29 So in this case, the probability will be equal to since he is stopping after 3 throats.

00:34 So first he will get 4, that is 1 upon 4.

00:37 Multiplyed by again 1 upon 4 because he is stopping at the third throw.

00:43 So for third throw, he can get 1 upon 4.

00:46 He can get 1 that is 1 upon 4, he can get 2 that is revivarity 1 upon 4 or he can get 3 that is 1 upon 4.

00:54 Or he can get 3 that is 1 upon 4 because all 3 have 1 upon 4 vibranity.

00:59 So from here we can write this is 1 upon 16 multiplied by 3 divided by 4.

01:04 That is 64 by, this is 3 by 64.

01:09 So this will be equal to 0 .0469.

01:12 This is the required probability.

01:14 Next we need to find out the expected value.

01:17 So x here is following geometric distribution with probability is equal to 3 by 4.

01:24 And expectation of, expectation of geometric is given by q divided by p.

01:33 So q here will be equal to 1 minus p, that is 1 .4, divided by 3 upon 4.

01:39 So this will be equal to 1 upon 3.

01:42 This is the expected value.

01:43 The 3 part, he will get $1 .5...

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