Probability of Landing Heads on Certain Toss/Summing to 1.You are tossing a coin and it has a probability of p to show heads on any given toss. You keep on tossing the coin until you see a heads. Let X represent the number of tosses until you see the first heads.1. Find the probability that X is odd.2. Find the probability that X is even, DO NOT USE QUESTION 1.

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Cremolinoer · Accepted Answer

Step 1  P  (  X  ≡  0    (  mod    2  )  )  =  p      ∑          n      ≥      0        (  1  −  p      )          2      n      +      1        =                    p        (        1        −        p        )                    1        −        (        1        −        p                  )          2                      P  (  X  ≡  1    (  mod    2  )  )  =  p      ∑          n      ≥      0        (  1  −  p      )          2      n        =            p              1        −        (        1        −        p                  )          2                    Step 2                    p        (        1        −        p        )                    1        −        (        1        −        p                  )          2                      +            p              1        −        (        1        −        p                  )          2                      =                    2        p        −                  p          2                            2        p        −                  p          2                      =  1

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