List the sample space of four coins tossedsimultainously. 1) define events A: Al

Question

List the sample space of four coins tossedsimultainously. 1) define events    A: All coins thesame                            B: precisely 1 head                            C: at least 2 heads      Evaluate P(A)+P(B)+P(C). Do theseevents from a partition? _______________________________________________________________________________ (b) Redefine A as 'all tails'. Do A, B, C now from apartition? ehat is P(A)+P(B)+P(C)? List the sample space of four coins tossedsimultainously. 1) define events    A: All coins thesame                            B: precisely 1 head                            C: at least 2 heads      Evaluate P(A)+P(B)+P(C). Do theseevents from a partition? _______________________________________________________________________________ (b) Redefine A as 'all tails'. Do A, B, C now from apartition? ehat is P(A)+P(B)+P(C)?

Explanation / Answer

The sample space would be: S = {HHHH, HHHT, HHTH, HHTT, TTTT, TTHH, TTHT, TTTH, HTTH,THHT, THTH, HTHT, HTHH, THTT, HTTT, THHH}   16possibilities A: All coins same ----->   HHHH, TTTT B: Precisely 1 head ---->   TTTH , HTTT,THTT, TTHT C: at least two heads ---> HHHH, HHHT, HHTH,HHTT, TTHH, HTTH, THHT, THTH, HTHT, HTHH, THHH 1) So, P(A) + P(B) + P(C) = 2/16 + 4/16 + 11/16 =17/16 Therefore theseevents do NOT form a partition because the sum of theirprobabilities do not equal 1. 2) Redefine A as all tails: A: All tails ----->   TTTT B: Precisely 1 head ---->   TTTH , HTTT,THTT, TTHT C: at least two heads ---> HHHH, HHHT, HHTH,HHTT, TTHH, HTTH, THHT, THTH, HTHT, HTHH, THHH A: All tails ----->   TTTT B: Precisely 1 head ---->   TTTH , HTTT,THTT, TTHT C: at least two heads ---> HHHH, HHHT, HHTH,HHTT, TTHH, HTTH, THHT, THTH, HTHT, HTHH, THHH P(A) + P(B) + P(C) = 1/16 + 4/16 + 11/16 = 16/16 = 1
Now these events doform a partition because they are now pair-wise disjoint and thesum of their probabilities equal one.

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