a coin is tossed three times. an outcome is represented by a string of the sort htt (meaning a head on the…
a coin is tossed three times. an outcome is represented by a string of the sort htt (meaning a head on the first toss, followed by two tails). the 8 outcomes are listed in the table below. note that each outcome has the same probability. for each of the three events in the table, check the outcome(s) that are contained in the event. then, in the last column, enter the probability of the event.\n\n| outcomes | probability |\n| ---- | ---- |\n| tth | |\n| thh | |\n| hht | |\n| ttt | |\n| hth | |\n| htt | |\n| hhh | |\n| tht | |\n\nevent a: a tail on both the first and the last tosses\n\nevent b: a head on each of the first two tosses\n\nevent c: exactly one head
Answer
Explanation:
Step1: Recall probability formula
The probability of an event $P(E)=\frac{n(E)}{n(S)}$, where $n(E)$ is the number of elements in the event $E$ and $n(S)$ is the number of elements in the sample - space. Here, $n(S) = 8$ since there are 8 possible outcomes when a coin is tossed 3 times.
Step2: Analyze Event A
Event A: A tail on both the first and the last tosses. The outcomes that satisfy this are TTH and TTT. So, $n(A)=2$. Then $P(A)=\frac{2}{8}=\frac{1}{4}$.
Step3: Analyze Event B
Event B: A head on each of the first two tosses. The outcomes that satisfy this are HHT and HHH. So, $n(B)=2$. Then $P(B)=\frac{2}{8}=\frac{1}{4}$.
Step4: Analyze Event C
Event C: Exactly one head. The outcomes that satisfy this are TTH, THT, and HTT. So, $n(C)=3$. Then $P(C)=\frac{3}{8}$.
| Event | TTH | THH | HHT | TTT | HTH | HTT | HHH | THT | Probability |
|---|---|---|---|---|---|---|---|---|---|
| Event A | $\checkmark$ | $\checkmark$ | $\frac{1}{4}$ | ||||||
| Event B | $\checkmark$ | $\checkmark$ | $\frac{1}{4}$ | ||||||
| Event C | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\frac{3}{8}$ |
Answer:
| Event | Probability |
|---|---|
| Event A | $\frac{1}{4}$ |
| Event B | $\frac{1}{4}$ |
| Event C | $\frac{3}{8}$ |