Flip a coin 14 times. If you get 6 tails or less, I will pay you $61$. Otherwise, you pay me $38.
Step 1 of 2 :
Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
SOLUTION :
P(tail) = P(not tail) = 0.5
=> p = q = 0.5
P(x ≤ 6 tails) = q^14 + C(14, 1) p q^13 + C(14, 2) p^2 q^12 + C(14,3) p^3q^11
+ C(14, 4) p^4 q^10 + C(14. 5)p^5 q ^9 + p^6 q^8
= 0.5^14 + 14*05^14 + 91*0.5^14 + 364*0,5^14 + 1001*0.5^14
+ 2002*0.5^14 + 3003*0.5^14
= 0,5^14 (1+ 14 + 91 + 364 + 1001 + 2002 + 3003)
= 0.5^6 (6476)
= 0.39526
=> P (x > 6) = 1 - P(x ≤ 6) = 1 - 0.39526 = 0.60474
Expected value of the proposition
= 61*0.39526 - 38*0.60474
= 1.13074 ($)
= 1.13 ($) (ANSWER).
P( getting 6 tails or less)=P(X<=6) =∑x=0x (nCx) px(1−p)(n-x) = 0.3953
expected value of the proposition = E(X) =expected win -expected lose
=61*P(gettting 6 or less tails) -38*(1-P(getting 6 tail or less)) =61*(0.3953)-38*(1-0.3953) = $ 1.13
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> In the 4th line, please read, " p^6 q^8 * as " C(14, 6) p^6 q^8 " .
Tulsiram Garg Sat, Sep 11, 2021 11:28 AM