In die and coin problems, unless stated otherwise, it is assumed coins and dice are fair and repeated trials are independent. Show Problem You purchase a certain product. The manual states that the lifetime $T$ of the product, defined as the amount of time (in years) the product works properly until it breaks down, satisfies $$P(T \geq t)=e^{-\frac{t}{5}}, \textrm{ for all } t \geq 0.$$ For example, the probability that the product lasts more than (or equal to) $2$ years is $P(T \geq 2)=e^{-\frac{2}{5}}=0.6703$. I purchase the product and use it for two years without any problems. What is the probability that it breaks down in the third year?
Problem You toss a fair coin three times:
Problem For three events $A$, $B$, and $C$, we know that
Find $P(A), P(B)$, and $P(C)$.
Problem Let $C_1, C_2,\cdots,C_M$ be a partition of the sample space $S$, and $A$ and $B$ be two events. Suppose we know that
Prove that $A$ and $B$ are independent.
Problem In my town, it's rainy one third of the days. Given that it is rainy, there will be heavy traffic with probability $\frac{1}{2}$, and given that it is not rainy, there will be heavy traffic with probability $\frac{1}{4}$. If it's rainy and there is heavy traffic, I arrive late for work with probability $\frac{1}{2}$. On the other hand, the probability of being late is reduced to $\frac{1}{8}$ if it is not rainy and there is no heavy traffic. In other situations (rainy and no traffic, not rainy and traffic) the probability of being late is $0.25$. You pick a random day.
Problem A box contains three coins: two regular coins and one fake two-headed coin ($P(H)=1$),
Problem Here is another variation of the family-with-two-children problem [1] [7]. A family has two children. We ask the father, "Do you have at least one daughter named Lilia?" He replies, "Yes!" What is the probability that both children are girls? In other words, we want to find the probability that both children are girls, given that the family has at least one daughter named Lilia. Here you can assume that if a child is a girl, her name will be Lilia with probability $\alpha \ll 1$ independently from other children's names. If the child is a boy, his name will not be Lilia. Compare your result with the second part of Example 1.18.
Problem
Problem Okay, another family-with-two-children problem. Just kidding! This problem has nothing to do with the two previous problems. I toss a coin repeatedly. The coin is unfair and $P(H)=p$. The game ends the first time that two consecutive heads ($HH$) or two consecutive tails ($TT$) are observed. I win if $HH$ is observed and lose if $TT$ is observed. For example if the outcome is $HTH\underline{TT}$, I lose. On the other hand, if the outcome is $THTHT\underline{HH}$, I win. Find the probability that I win.
The print version of the book is available through Amazon here. What is the probability of getting 2 tails when a coin is tossed 3 times?Probability of obtaining 2 heads or 2 tails is =86=43.
What is the probability of getting 2 consecutive heads when a coin is tossed 3 times?Answer: If you flip a coin 3 times, the probability of getting at least 2 heads is 1/2. Let's look into the possible outcomes.
When a coin is tossed 3 times what is the probability it will result in 2 heads and 1 tail?What is the probability of two heads and one tail? Summary: The Probability of getting two heads and one tails in the toss of three coins simultaneously is 3/8 or 0.375.
What is the probability of getting 2 tails?For example, the probability of two heads is 1/2 · 1/2 = 1/4, and the probability of two tails is the same. Since there are two leaves corresponding to one head and one tail, each of probability 1/4, the probability of this event is 1/4 + 1/4 = 1/2.
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