This Exam P sample reference tests Convolution. There are eight feasible weekly splits of seven claims, and every convolution term equals 2⁻⁹. Their sum is 1/64, choice D.
Condition on the first week's count n. For a total of seven, the second week's count must be 7-n, with n ranging from 0 through 7.
Pr(N1+N2=7)=n=0∑7Pr(N1=n)Pr(N2=7−n)
Model
Model
Independence gives a discrete convolution. Under the supplied mass function, every one of the eight feasible split probabilities simplifies to 2 to the power -9.
Pr(N1=n)Pr(N2=7−n)=2−(n+1)2−(8−n)=2−9
Compute
Compute
Summing the eight equal convolution terms gives 8 × 2 to the power -9, or 2 to the power -6.
Pr(N1+N2=7)=8⋅2−9=2−6
Answer
Answer
Therefore the probability of seven claims in total is 1/64, corresponding to choice D.
641(D)
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AThe value 1/256 equals two of the eight convolution terms, so it counts only two feasible weekly splits.
BThe value 1/128 equals four of the eight equal terms and therefore counts only half of the feasible splits.
CThe value 7/512 counts seven split values. The endpoints n=0 and n=7 are both feasible, giving eight terms.
EThe value 1/32 equals sixteen terms of size 1/512, double the actual number of feasible weekly splits.
Original practice · fully worked
Original variant: convolution of two uniform counters
Two independent counters each display one of the integers 0 through 5 with equal probability. Determine the probability that their displayed values sum to 7.
A 1/18
B 1/12
C 5/36
D 1/6
E 1/9
Variant answer in brief
The feasible ordered pairs are (2,5), (3,4), (4,3), and (5,2). Four of 36 equally likely pairs give 1/9, choice E.
Setup
Setup
The ordered pair of counter values is uniformly distributed over 36 pairs in the six-by-six sample space.
(X,Y)∈{0,1,2,3,4,5}2
Model
Model
Enumerate the ordered pairs whose coordinates sum to seven while both coordinates remain between zero and five.
X+Y=7⟹(X,Y)∈{(2,5),(3,4),(4,3),(5,2)}
Compute
Compute
There are four favorable ordered pairs, each with probability 1/36, so the total probability is 4/36=1/9.
Pr(X+Y=7)=4(61)2=91
Answer
Answer
The probability that the displayed values sum to seven is 1/9, which is choice E.
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