Suppose I have either a fair coin or a bent coin, and I don’t know which. The bent coin has a 60 % 60\% 60% probability of coming up heads. I throw the coin ten times and it comes up heads 8 times. What is the probability I have the fair coin vs. the probability I have the bent coin? Assume at the outset there is an equal ( . 5 , . 5 ) (.5,.5) (.5,.5) prior probability of either coin.

the Probability of Fair coin = 0.2664=26.64 % and

Probability of Bent coin = 0.7336= 73.36 %

Step by step Explanation:

The fair coin has a 60% probability of coming up heads, same thing with the bent coin , when the coin was thrown up ten times,it comes up heads 8 times,

Here we were told to find the probability fair coin vs. probability bent coin

A)THE FAIR COIN::

The probability of getting Head Coin = 1/2

Probability of getting tail Coin = 1/2

Then the Probability of head 8 times out of 10 can be calculated using combination

= ¹⁰C₈ × (1/2)⁸ ×(1/2)²

= 45/1024

= 0.0439

Therefore, the Probability of head 8 times out of 10= 0.0439

B)BENT COIN

the probability of getting Head Coin = 0.6

= 6/10

= 3/5

NOTE: 60%=0.6

The probability of getting tail Coin = 1 - 3/5

= 2/5

Then, the probability of getting head 8 times out of 10 can be calculated using combination;

= ¹⁰C₈ × (3/5)⁸ × (2/5)²

= 45 × 6561 ×( 4 / 5¹⁰)

= 0.1209

Therefore, the probability of getting head 8 times out of 10 = 0.1209

We can now calculate the Probability of Fair coin

(Probability of head 8 times out of 10 in fair coin )/(Probability of head 8 times out of 10 in fair coin )+(the probability of getting head 8 times out of 10 in bent coin)

= (0.0439 )/(0.0439 + 0.1209)

= 0.2664

Probability of bent coin can as well be calculated as 0.1209 /(0.0439 + 0.1209)

= 0.7336

Therefore, the Probability of Fair coin = 0.2664=26.64 % and

Probability of Bent coin = 0.7336= 73.36 %

0 is its x and y coordinate