Question

Three fair dice (black, white and red) are tossed simultaneously and the value recorded as an ordered triple y = (yb, yw, yr), which is a point in S.

Let A1, A2, A3 be the events A1 = {(yb, yw, yr) : yb = yw} A2 = {(yb, yw, yr) : yb = yr} A3 = {(yb, yw, yr) : yw = yr}

(i) What is the sample space S for this experiment? How many points or elementary events does it contain?

(ii) How many points are contained in the event A1?

(iii) How many points are contained in the event A1 ∪ A2?

(iv) How many points are contained in the event A1 ∪ A2 ∪ A3?

(v) How many points are contained in the event A1 ∩ A2?

(vi) How many points are contained in the event A1 ∩ A2 ∩ A3?

(vii) The events A1 and A2 ∩ A3 are disjoint. (True or False?)

(viii) The events A1 and A2 are independent. (True or False?)

(viii) The events A1 and A3 are independent. (True or False?)

(ix) The events A2 and A3 are independent. (True or False?)

(x) The events A1, A2 and A3 are independent. (True or False?)

Answer 1

pl note that i have used simple counting and multiplication rule in parts 3,4. For finding number of elements of (i,i,j) note that we have 6 options to choose i, then 1 option to choose the same i as first, and then 5 options to choose j such that j and i are not equal. Also note that while taking union, we consider the intersection term, that is common to both sets, twice, thus we subtract it once. In part 4, we need to use: |A1 U A2U A3| = |A1| + |A2| +|A3| -|A1 u A2| -|A1 U A3| - |A2 U A3| + |A1 U A2 U A3|

And intersections in 5, 6 are same.

For any doubts please comment and ask

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