Original practice · fully worked
Original variant: access digits under a mismatch condition
An authentication tester independently generates a first digit uniformly from 1, 2, 3, 4, and 5 and a second digit uniformly from 1, 2, 3, and 4. Given that the two generated digits are different, calculate the probability that their sum is 5.
- A
1/5 - B
1/4 - C
4/15 - D
4/9 - E
3/4
Answer and calculation
There are 20 equally likely ordered digit pairs, and removing the four equal pairs leaves 16 under the condition. Four of those pairs sum to 5, so the conditional probability is 1/4 and choice B.
Setup
Setup
Represent the generated digits by an ordered pair. The unequal ranges produce 5 × 4 equally likely pairs.
#{(X,Y)}=5(4)=20 Model
Model
The condition removes the four possible equal pairs; a pair with two fives is unavailable because the second digit cannot be five.
#{X=Y}=20−4=16 Compute
Compute
Enumerate the sum-five pairs. Every one of them also satisfies the condition.
{(X,Y):X+Y=5}={(1,4),(2,3),(3,2),(4,1)} Pr(X+Y=5∣X=Y)=164=41 Answer
Answer
The required conditional probability is 1/4.
41(B) 