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Solutions to probability-related homework problems for math 102 and core 143. The scenarios include picking hats with different colors, pulling socks from a drawer, and rolling dice. The calculations involve using the not rule and the multiplication principle.
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Math 102 & Core 143
a) Katie doesn’t feel like wearing a black hat today, so given that she picked a black hat first, what is the chance that she picks a white hat in the next two picks?
We use the not rule (we don’t want both of the next two picks each to be either black or blue). 1 − 69 · 58 = 1 − 1024 = 127
b) What is the chance that in the first three picks, at least one of the hats is not black?
We use the not rule (we don’t want all three picks to be black). 1 − 105 · 49 · 38 = 1 − 121 = 1112.
a) What is the chance that the first two socks pulled out are the black pair? 102 · 19 = 451.
b) Suppose Fred pulls out the first sock, but drops it under the bed before noticing the color. He quickly pulls out another sock. What is the chance that the second sock is an odd sock?
There is no new information here. Think of the chance that the second card in a deck is an ace. It is 524 not 514. If we threw the first card under the dresser, it would still be 524. Hence, the answer here is 104.
c) If the first sock pulled out is white, what is the chance that he gets a white pair at or before the fourth sock?
We use the not rule. We don’t want the second, third, and fourth socks all to be “not white.” Hence, we have 1 − 69 · 58 · 47 = 1 − 1042 = 1924.
d) What is the chance that the first two socks are a pair?
The pair could be either black or white. Hence, we have 102 · 19 + 104 · 39 = 457.
This is the multiplication principle. Hence, the answer is
( 1 5
) (^8) .
This is the Not Rule again. Using our answer to (a), we see that the probability of not getting at
( 25 36
) (^10)