Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.
Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.
Let $L_{1},\, L_{2}, L_{3}$ be three letters and $E_{1},\, E_{2},$ and $E_{3}$ be their corresponding envelops respectively.
There are $6$ ways of inserting $3$ letters in $3$ envelops. These are as follows:
$\left.\begin{array}{l}L_{1} E_{1}, \,L_{2} E_{3},\, L_{3} E_{2} \\ L_{2} E_{2}, \,L_{1} E_{3}, \,L_{3} E_{1} \\ L_{3} E_{3}, \,L_{1} E_{2},\, L_{2} E_{1} \\ L_{1} E_{1},\, L_{2} E_{2},\, L_{3} E_{3}\end{array}\right]$
$L_{1} E_{2}, \,L_{2} E_{3},\, L_{3} E_{1}$
$L_{1} E_{3},\, L_{2} E_{1},\, L_{3} E_{2}$
There are $4$ ways in which at least one letter is inserted in a proper envelope.
Thus, the required probability is $\frac{4}{6}=\frac{2}{3}$
Similar Questions
$A$ and $B$ toss a coin alternatively, the first to show a head being the winner. If $A$ starts the game, the chance of his winning is
$A$ and $B$ toss a coin alternatively, the first to show a head being the winner. If $A$ starts the game, the chance of his winning is
A card is drawn at random from a pack of $52$ cards. The probability that the drawn card is a court card i.e. a jack, a queen or a king, is
A card is drawn at random from a pack of $52$ cards. The probability that the drawn card is a court card i.e. a jack, a queen or a king, is
The probability of getting a number greater than $2$ in throwing a die is
The probability of getting a number greater than $2$ in throwing a die is
Two dice are thrown. The events $A, B$ and $C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\leq 5$
Describe the events $A$ or $B$
Two dice are thrown. The events $A, B$ and $C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\leq 5$
Describe the events $A$ or $B$
From a pack of $52$ cards two cards are drawn in succession one by one without replacement. The probability that both are aces is
From a pack of $52$ cards two cards are drawn in succession one by one without replacement. The probability that both are aces is