The probabilities that players $A$ and $B$ of a team are selected for the captaincy for a tournament are $0.6$ and $0.4$,respectively. If $A$ is selected as the captain,the probability that the team wins the tournament is $0.8$ and if $B$ is selected as the captain,the probability that the team wins the tournament is $0.7$. Then the probability,that the team wins the tournament,is:
- A$0.74$
- B$0.76$
- C$0.72$
- D$0.78$
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$A$ boy throws an unbiased die. Whenever he gets $1$ on the die,he has a further chance to throw it once again immediately. The probability that the boy gets a score of $7$ in this process is
$A$ coin is tossed $m + n$ times,where $m \ge n.$ The probability of getting at least $m$ consecutive heads is
Difficult
View SolutionLet a computer program generate only the digits $0$ and $1$ to form a string of binary numbers. The probability of occurrence of $0$ at even places is $\frac{1}{2}$ and the probability of occurrence of $0$ at odd places is $\frac{1}{3}$. Then the probability that $'10'$ is followed by $'01'$ is equal to:
DifficultJEE MAIN 2021
View SolutionLet $\alpha$ be a root of $x^2+x+1=0$ and suppose that a fair die is thrown $3$ times. If $a, b,$ and $c$ are the numbers shown on the die,then the probability that $\alpha^a+\alpha^b+\alpha^c=0$ is
Two players,$P_1$ and $P_2$,play a game against each other. In every round,each player rolls a fair die once. Let $x$ and $y$ denote the outcomes for $P_1$ and $P_2$. If $x > y$,$P_1$ scores $5$ points and $P_2$ scores $0$. If $x = y$,each scores $2$ points. If $x < y$,$P_1$ scores $0$ and $P_2$ scores $5$. Let $X_n$ and $Y_n$ be the total scores of $P_1$ and $P_2$ after $n$ rounds. Match the following:
List-$I$ List-$II$ $(I)$ Probability of $(X_2 \geq Y_2)$ is $(P)$ $\frac{3}{8}$ $(II)$ Probability of $(X_2 > Y_2)$ is $(Q)$ $\frac{11}{16}$ $(III)$ Probability of $(X_3 = Y_3)$ is $(R)$ $\frac{5}{16}$ $(IV)$ Probability of $(X_3 > Y_3)$ is $(S)$ $\frac{355}{864}$ $(T)$ $\frac{77}{432}$
| List-$I$ | List-$II$ |
| $(I)$ Probability of $(X_2 \geq Y_2)$ is | $(P)$ $\frac{3}{8}$ |
| $(II)$ Probability of $(X_2 > Y_2)$ is | $(Q)$ $\frac{11}{16}$ |
| $(III)$ Probability of $(X_3 = Y_3)$ is | $(R)$ $\frac{5}{16}$ |
| $(IV)$ Probability of $(X_3 > Y_3)$ is | $(S)$ $\frac{355}{864}$ |
| $(T)$ $\frac{77}{432}$ |
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