PROBABILITY PART 5-_-?

1.
In 52 playing cards there are 13 clubs and 13 hearts.
P(1st card is a club or heart)
= 26/52
= 1/2

In 52 playing cards there are 4 suits. Thus, there are 12 face cards (3 × 4).
P(2nd card is a face cards)
= 12/52
= 3/13

P([1st card is a club or heart] and [2nd card is a face card])
= P(1st card is a club or heart) × P(2nd card is a face card)
= (1/2) × (3/13)
= 3/26


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2.
(a)(a)
P(pick a teacher)
= P(room B and pick a teacher)
= (1/3) × (1/2)
= 1/6

(a)(b)
P(pick a student)
= P([room A and pick a student] or [room B and pick a student])
= (1/3) × (2/3) + (1/3) × (1/2)
= (2/9) + (1/6)
= (4/18) + (3/18)
= 7/18

(a)(c)
P(pick a social worker)
= P([room A and pick a social worker] or [room C and pick a social worker])
= (1/3) × (1/3) + (1/3) × 1
= (1/9) + (3/9)
= 4/9

(b)(i)
P(both pick a teacher)
= (1/6)²
= 1/36

(b)(ii)
P(both pick a social worker in the same room)
= P([both pick a social worker in room A] or [both pick a social worker in room C])
= [(1/3) × (1/3)]² + [(1/3) × 1]²
= (1/81) + (1/9)
= (1/81) + (9/81)
= 10/81


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3.
W : Vincent wins a game.
L : Vincent loses a game.

P(W) = 0.6
P(L) = 1 - 0.6 = 0.4

P(Vincent wins the match)
= P(WW) + P(WLW) + P(LWW)
= 0.6 × 0.6 + 0.6 × 0.4 × 0.6 + 0.4 × 0.6 × 0.6
= 0.36 + 0.144 + 0.144
= 0.648

P(Vincent wins a one-game match)
= 0.6

The above game rule is more advantageous to Vincent because of the greater probability.