Question

S = 0000100001000110000001010000010010000000000000000010000000010000110010001000100001011010100000010100; /*1 = hit; 0 = miss*/

Compute and report the following for your sequence of 100 shots: 1. P(S_k=1 | S_k-1=1), the probability of a hit immediately following a hit. 2. P(S_k=1 | S_k-1=0), the probability of a hit immediately following a miss. 3. P(S_k=1 | S_k-1=1) - P(S_k=1 | S_k-1=0), the difference of the probabilities from 1 and 2. This will be your test statistic.

Answer 1

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SOLUTION:

a) count all 1s then, count such 1s which also had 1s just after them.

P(S_k=1 | S_k-1=1)

= n(S_k=1 S_k-1=1) / n(S_k-1=1)

= 3/22

=0.1364

a) count all 0s(except the one in last position as it has no follower i.e. next element) then, count all 0s which had 1s just after them.

P(S_k=1 | S_k-1=0)

= n(S_k=1 S_k-1=0) / n(S_k-1=0)

= 19/77

= 0.2468

P(S_k=1 | S_k-1=1) - P(S_k=1 | S_k-1=0)

= 3/22 - 19/77

= -17/154

= -0.1104

It is a negative value.