Q. 5.7 - A First Course in Probability

Question

To be a winner in a certain game, you must be successful in three successive rounds. The game depends on the value of U, a uniform random variable on $(0,1)$. If $U>.1$, then you are successful in round $1$; if $U>1$, then you are successful in round $2$; and if $U>3$, then you are successful in round $3$.

 (a) Find the probability that you are successful in round $1$.

 (b) Find the conditional probability that you are successful in round $2$ given that you were successful in round $1$.

 (c) Find the conditional probability that you are successful in round $3$ given that you were successful in rounds $1 and 2$ 

 (d) Find the probability that you are a winner 

Step-by-Step Solution

Answer

(a)  The probability are success in round $1$ is $0.9$

(b) The conditional probability are success in round $2$ is $8/9$

(c) The conditional probability are success in round $3$ is $7/8$

(d) The winning  of the is  probability $0.7$

Step: 1 Round 1 (part a)

We are successful in round one if and therefore only if $U>0.1$. Each occurrence has a possibility of occurring.

$$\begin{gathered} P(U>0.1) \\ =1-0.1 \\ =0.9 \end{gathered}$$

Step:2 Round 2 (part b)

(b) According to our sources, we were successful in the first round., i.e. $u>0.1$ We're seeking for the conditional probability of winning the round. 2, i.e. $u>0.2$

However, this is straightforward since

$$\begin{gathered} P(U>0.2\mid U>0.1) \\ =\frac{P(U>0.2,U>0.1)}{P(U>0.1)} \\ =\frac{P(U>0.2)}{P(U>0.1)} \\ =\frac{0.8}{0.9} \\ =\frac{8}{9} \end{gathered}$$

Step:3 Round 3 (part c)

(c) We were notified that we were successful in both the first and second rounds, i.e., $U=0.1\text{ and }U=0.2$. The conditional likelihood that we'll win round three, i.e. $U>0.3$, is what we're looking for. This, on the other hand, is simple since

$$\begin{gathered} P(U>0.3\mid U>0.1,U>0.2) \\ =\frac{P(U>0.3,U>0.2,U>0.1)}{P(U>0.1,U>0.2)} \\ =\frac{P(U>0.3)}{P(U>0.2)} \\ =\frac{0.7}{0.8} \\ =\frac{7}{8} \end{gathered}$$

Step :4 Probability winner (part d)

(d) We are the victors if and only if we pass all three rounds. $U>0.3$ is the frequency of the phenomenon. That event has a chance of happening.

$$\begin{gathered} P(U>0.3) \\ =1-0.3 \\ =0.7 \end{gathered}$$