In a London State daily lottery, a sequence of 3 (not necessarily different) digits in the range 0-7 are selected at random. Find the probability that 2 of 3 digits are the same.
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 35 } { 384 }$
D) $\frac { 7 } { 256 }$
E) $\frac { 21 } { 64 }$

Question

In a London State daily lottery, a sequence of 3 (not necessarily different) digits in the range 0-7 are selected at random. Find the probability that 2 of 3 digits are the same.  
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 35 } { 384 }$
D) $\frac { 7 } { 256 }$
E) $\frac { 21 } { 64 }$

Quizplus · Accepted Answer

To find the probability that exactly 2 of the 3 digits are the same in a sequence where each digit can range from 0 to 7, we can use combinatorial methods. There are $8^3 = 512$ total possible sequences since each digit can be one of 8 numbers (0 through 7), and there are 3 digits.For exactly 2 of the 3 digits to be the same, we can break it down into steps:1. Choose which digit will be repeated (8 choices, since there are 8 possible digits).2. Choose which 2 positions out of the 3 will have this repeated digit ($C(3, 2) = 3$ ways).3. Choose a different digit for the remaining position (7 choices, since it can't be the same as the repeated digit).Multiplying these choices gives us the total number of favorable outcomes: $8 * 3 * 7 = 168$.The probability is then the number of favorable outcomes divided by the total number of outcomes: $\frac{168}{512} = \frac{21}{64}$.

In a London State Daily Lottery, a Sequence of 3 12\frac { 1 } { 2 }21​

In a London State Daily Lottery, a Sequence of 3 $\frac { 1 } { 2 }$