Question

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

Answer 1

(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)