Answer :
The needed probability to achieve at least 5 tails is =1
What is meant by Probability?
Probability: Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.
A should be the occurrence of at least 5 tails. Then Ac is the eventuality of receiving a maximum of 5 tails. Thus, P(A)+P([tex]A^c[/tex])=1
This problem can be resolved explicitly by following the steps below. The probability that x tails will fall when a coin is tossed n times is
C(n, x)[tex](1/2)^x[/tex][tex](1/2)^n[/tex]=C(n, x)[tex](1/2)^n[/tex]
Assume that A gets at least 5 tails and B gets a maximum of 5 tails. Therefore, the necessary probability will be
P(A∪B)=P(A)+P(B)-P(A∩B)
Now, getting 5 or more tails equates to getting at least 5 tails. Therefore, P(A)=∑8x=5 C(8,x) (1/2)^8
=C(8,5)(1/2)^8+C(8,6)(1/2)^8+C(8,7)(1/2)^8 +C(8,8)(1/2)^8
=[C(8,5)+C(8,6)+C(8,7)+C(8,8)]/2^8=(56+28+8+1)/256=93/256
Once more, having 5 tails or less means getting no more than 5.
P(B) =∑5x=0 C (8, x) (1/2)^8
=[C(8,0)+C(8,1)+C(8,2)+C(8,3)+C(8,4)+C(8,5)]/256
=(1+8+28+56+70+56)/256=219/256
Finally, A∩B receiving precisely 5 tails. P(A∩B)=C(8,5)(1/2)^8=56/256 as a result.
The needed probability is therefore (93+219-56)/256=256/256=1
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