20. At Inglls Park in Owen Sound, you can see adult salmon jumping over a series of logs as they swim upstream to spawn. The salmon have a 0.6 probability of successful jump if they rest prior to the jump, but jumping only a 0.3 probability immediately after jumping the previous logs. If the fish are rested when they come to the first log, what is the probability that a salmon will dear a) both of the first two logs on the first try without resting? b) all of the first four logs on the first try if it rests after the second jump?
Solution
Step 1 ：Given the probability of a successful jump after resting is 0.6 and without resting is 0.3.
Step 2 ：For part a), we need to find the probability of the salmon clearing the first log and then immediately clearing the second log without resting. Since the events are independent, we can multiply the probabilities: \(0.3 \times 0.3 = 0.09\)
Step 3 ：For part b), we need to find the probability of the salmon clearing the first two logs without resting (which we already calculated in part a)), then resting, and then clearing the next two logs. Since the events are independent, we can multiply the probabilities: \(0.09 \times 0.6 \times 0.6 = 0.0324\)
Step 4 ：\(\boxed{\text{a) The probability that a salmon will clear both of the first two logs on the first try without resting is 0.09.}}\)
Step 5 ：\(\boxed{\text{b) The probability that a salmon will clear all of the first four logs on the first try if it rests after the second jump is 0.0324.}}\)
