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A, B, and C take turns flipping a coin. The first one to get a head wins. The sample space of this experiment can be defined by

$S=\left\{\begin{array}{l}1,01,001,0001,...,\\ 0000···\end{array}\right.$

(a) Interpret the sample space.

(b) Define the following events in terms of S:

(i) $A$wins = $A$.

(ii) $B$wins = $B$.

(iii) ${(A∪B)}^c$.

Assume that A flips first, then B, then C, then A, and so on.

A town contains $4$people who repair televisions. If$4$sets break down, what is the probability that exactly$i$of the repairers is called? Solve the problem for$i=1,2,3,4.$What assumptions are you making?

Let $A$denote the event that the midtown temperature in Los Angeles is${70}^{â—¦}F$, and let$B$denote the event that the midtown temperature in New York is${70}^{â—¦}F$. Also, let's $C$denote the event that the maximum of the midtown temperatures in New York and in Los Angeles is${70}^{â—¦}F$. If $P\left(A\right)=.3,P\left(B\right)=.4,$and$P\left(C\right)=.2$, find the probability that the minimum of the two midtown temperatures is${70}^{â—¦}F$.

$\underset{1}{\overset{∞}{∪}}{E}_{i}F=\left(\underset{1}{\overset{∞}{∪}}{E}_i\right)F$and$\underset{1}{\overset{∞}{âˆ\copyright }}\left(E_i∪F\right)=\left(\underset{1}{\overset{∞}{âˆ\copyright }}{E}_i\right)∪F$

If a die is rolled 4 times, what is the probability that 6 comes up at least once?

If a die is rolled $4$times, what is the probability that $6$

comes up at least once?

Two dice are thrown $n$times in succession. Compute

the probability that a double $6$appears at least once. How large need $n$be to make this probability at least$\frac{1}{2}$?

$\left(a\right)$If $N$people, including $A$and$B$, are randomly arranged in a line, what is the probability that $A$and $B$are next to each other?

$\left(b\right)$What would the probability be if the people were randomly arranged in a circle?

How many people have to be in a room in order that the probability that at least two of them celebrate their birthday in the same month is at least $\frac{1}{2}?$ Assume that all possible monthly outcomes are equally likely.

If there are $12$strangers in a room, what is the probability that no two of them celebrate their birthday in the same month?