SOLUTION: You are given an unfair coin with probability of obtaining a head equal to 1/3, 000, 000, 000. You toss this coin 6,000,000,000 times. Let A be the event that you get “tails fo

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Question 1198328: You are given an unfair coin with probability of obtaining a head equal to
1/3, 000, 000, 000. You toss this coin 6,000,000,000 times. Let A be the
event that you get “tails for all the tosses”. Let B be the event that you get
“heads for all the tosses”.
i) Approximate P (A).
ii) Approximate P (A ∪ B)
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
Let be the number of heads among the tosses. Then, ∼ (,).
given:
probability of obtaining a head equal to:
You toss this coin times

Here, we have small and large . So, we can apply Poisson approximation.
In other words, (,) is well-approximated by where .

(a)
= [] = ≈
(b)
( ∪ )= []+ [] =
The second term is compared to the first one.
Hence, ( ∪ ) is approximately the same as .

Answer by ikleyn(52818)   (Show Source): You can put this solution on YOUR website!
.
You are given an unfair coin with probability of obtaining a head equal to 1/3, 000,000,000.
You toss this coin 6,000,000,000 times. Let A be the event that you get “tails for all the tosses”.
Let B be the event that you get “heads for all the tosses”.
i) Approximate P (A).
ii) Approximate P (A ∪ B)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

For (i),  P(A) =  = ,

          where n = 3,000,000,000.


          From Calculus, it is well known fact that  

               lim  =   as n --> oo.


          So, P(A) =  =  = 0.1353  (approximately).    ANSWER



For (ii), P(B) is extremely small and events A and B are mutially exclusive;

          therefore, P(A U B) = P(A): P(B) contributes practically nothing.

Solved.



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