Math 115 Exam 2

Answers 1. A fair coin is tossed 7 times. What is the probability that there are exactly 3 heads. ( ) 3 4 1 1 7! 7 6 7,3 2 2 3!4! C ⋅ = = ⋅ 5 4321 ⋅ ⋅ ⋅ ⋅ ⋅ 321 ⋅ ⋅ 4321 ⋅ ⋅ ⋅ ⋅ 7 35 128 1 2 = 2. A person draws two socks at random out of a drawer containing 3 black socks and 4 red socks. Given that the second sock drawn is red what is the probability that the first sock is black. ( ) ( ) ( ) 3 4 7 6 3 3 4 4 7 6 7 6 1 and 2 1 2 2 1 2 PB R PB R PR ⋅ = = = ⋅ + ⋅ 3. A box contains ten apples, three of which are bruised. If seven apples are chosen at random from the box what is the probability that at least two of the chosen apples are bruised? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 7,7 7,6 3,1 at least 2 1 1 10,7 10,7 10 49 6 9 10 0 ,7 C C C P Pnone Pone C C C = − + = − + − − + = = = ⋅ = 3 8 ⋅ 4 3 2 ⋅ 120 1 = ⋅ 4. There are two dice numbered 1,1,2,2,3,3. The two dice are tossed. Let X be the smaller of the two numbers showing. What is the expected value of X? MIN 1 1 2 2 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2