In: Statistics and Probability

Assume we roll a fair four-sided die marked with 1, 2, 3 and
4.

(a) Find the probability that the outcome 1 is first observed after
5 rolls.

(b) Find the expected number of rolls until outcomes 1 and 2 are
both observed.

(c) Find the expected number of rolls until the outcome 3 is
observed three times.

(d) Find the probability that the outcome 3 is observed exactly
three times in 10 rolls

given that it is first observed after 5 rolls.

(e) Find the probability that the outcome 3 is first observed after
5 rolls given that

it is observed exactly three times in 10 rolls

(a) probability that the outcome 1 is first observed after 5 rolls.

P(1) = 1/4

P(not 1) = 1-1/4 = 3/4

probability(outcome 1 is first observed after 5 rolls)

= P(1 doesn't come in 5 rolls)

= P(not 1)^5

= (3/4)^5

= **0.2373**

(b) expected number of rolls until outcomes 1 and 2 are both observed.

The first number (either 1 or 2) will come up with probability 2/4 on each roll, and the number of rolls needed follows a geometric distribution, so 4/2 = 2 rolls are expected.

Then the remaining number comes up with probability 1/4, and similar to the first number, the reciprocal number of rolls are expected, or 4/1 = 4 rolls.

total rolls = 2+4 = 6

**Thus 6 rolls are expected on average.**

(c) expected number of rolls until the outcome 3 is observed three
times.

expected no. of rolls for event to occur x times = x / (p(event))

P(3) = 1/4

expected no. of rolls for outcome 3 to occur 3 times = 3 / (1/4)
= **12**

**expected no. of rolls for outcome 3 to occur 3 times =
12**

(d) probability that the outcome 3 is observed exactly three times in 10 rolls given that it is first observed after 5 rolls.

x=no. of 3

p=1/4

P(3 observed 3 times in 10 rolls | 3 not observed in first 5 rolls)

= P(3 observed 3 times in the 5 rolls afterr first 5 rolls)

= 5C3 * (1/4)^3 * (1 - 1/4)^(5-3)

= **0.0879**

(e) probability that the outcome 3 is first observed after 5 rolls
given that it is observed exactly three times in 10 rolls

x=no. of 3

p=1/4

P(3 observed 3 times in 10 rolls) = 10C3 * (1/4)^3 * (1 - (1/4))^(10-3)

= **0.2503**

P(3 is first observed after 5 rolls)

= P(3 doesn't come in 5 rolls)

= P(not 3)^5

= (1 - 1/4)^5

= **0.2373**

P(3 observed 3 times in 10 rolls | 3 is first observed after 5 rolls)

= **0.0879 {from previous part}**

P( 3 is first observed after 5 rolls | 3 observed 3 times in 10 rolls)

= P(3 observed 3 times in 10 rolls | 3 is first observed after 5 rolls)*P(3 is first observed after 5 rolls) / P(3 observed 3 times in 10 rolls)

= 0.0879 * 0.2373 / 0.2503

= 0.0833

**P( 3 is first observed after 5 rolls | 3 observed 3
times in 10 rolls) = 0.0833**

**(please UPVOTE)**

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided
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a. What is the pdf of W?
b What is E(W)?

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use R or draw the pmf...

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17#13
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Greater than 4
. Write your answer as a fraction or whole number.
Assume that a fair die is rolled. The sample space is
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