One side of a 6-sided die is marked \"0 \", two sides are marked \"2\", and thre

Question

One side of a 6-sided die is marked "0 ", two sides are marked "2", and three sides are marked "3". Each side is equally likely to occur when the die is rolled. The die is rolled twice and the results are recorded. Let A be the event that the first roll is 0. Let B be the event that the second roll is 3. Let C be the event that the sum of the two rolls is 5. (a) Find Pr(A), Pr(B), and Pr(C). (b) Find Pr(A Intersection B). Pr(A Intersection C), and Pr(B Intersection C). (c) Are A and C independent events? Are B and C independent events? (d) Find Pr(A Union C) and Pr(B Union C). (e) Find Pr(B Union C)). Blofeld captures James Bond and places him in a pit with 100 deadly scorpions, 60 of which are male and 10 of which are female. The male scorpions' bites are fatal 70% of the time and the female scorpions' bites are fatal 90% of the time. Bond escapes the pit, but is bitten once by one of the scorpions. Given that Bond survives, what is the probability that the scorpion that bit him was male? (Assume the scorpion that hit Bond was chosen uniformly at random from the 100 in the pit.) A string consisting of As, Bs and Cs is chosen uniformly at random from the set {BBBBB, ABBBC, AACCC, ABBCC, BBBBC}. Let X be the number of As in the string. Give the probability distribution of X.

Explanation / Answer

1.There are total 36 (6*6) cases if the dice rolled twice

a.

P(A) =6/36=1/6
first roll is 0 means we have 6 cases out of 36 cases
(0,0)(0,2)(0,2)(0,3)(0,3)(0,3)

P(B)=18/36=1/2
for one 0 face there are three times 3 in 2nd roll
so for one 0 face two 2 face three 3 face total 18 cases

P(C)= 12/36=1/3
for each 2 three cases (2,3)(2,3)(2,3) so total 6 cases
for each 3 two cases (3,2)(3,2) so total 6 cases
so total =6+6=12 cases

b.
P(A&B)=3/36=1/12
three cases (0,3)(0,3)(0,3)

P(A&C)=0
if first roll 0 sum cant be 5

P(B&C)=6/36=1/6
for each 2 three cases (2,3)(2,3)(2,3) so total 6 cases

c.A and C independent,if first roll 0 sum cant be 5
B and C not independent,for each 2 three cases (2,3)(2,3)(2,3) so total 6 cases

d.

P(AorB)=21/36
for B
one 0 face there are three times 3 in 2nd roll
so for one 0 face two 2 face three 3 face total 18 cases
and for A
3 extra (0,0)(0,2)(0,2)
so total =18+3=21 cases

P(BorC)=24/36=2/3

for B
one 0 face there are three times 3 in 2nd roll
so for one 0 face two 2 face three 3 face total 18 cases
for C
for each 3 two cases (3,2)(3,2) so total 6 cases more
so total = 6+18= 24

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