Abag contains 3 coins of which one is two headed and the other two are normal & fair. A coin is selected
a random and tossed 4 times in succession. If all the four times it appears to be head what is the probability
that the two headed coin was selected.
Answers
Answer:
0.44
Step-by-step explanation:
Let E
1
,E
2
, and E
3
be the respective events of choosing a two headed coin, a biased coin, and an unbiased coin.
∴P(E
1
)=P(E
2
)=P(E
3
)=
3
1
Let A be the event that the coin shows heads.
A two-headed coin will always show heads.
∴P(A∣E
1
)=P(coinshowingheads,giventhatitisatwo−headedcoin)=1
Probability of heads coming up, given that it is a biased coin =75%
∴P(A∣E
2
)=P(coinshowingheads,giventhatitisabiasedcoin)=
100
75
=
4
3
Since the third coin is unbiased, the probability that it shows heads is always
2
1
∴P(A∣E
3
)=P(coinshowingheads,giventhatitisanunbiasedcoin)=
2
1
The probability that the coin is two-headed, given that it shows heads, is given by P(E
1
∣A).
By using Baye's theorem, we obtain
P(E
1
∣A)=
P(E
1
)⋅P(A∣E
1
)+P(E
2
)⋅P(A∣E
2
)+P(E
3
)⋅P(A∣E
3
)
P(E
1
)⋅P(A∣E
1
)
=
3
1
⋅1+
3
1
⋅
4
3
+
3
1
⋅
2
1
3
1
⋅1
=
3
1
(1+
4
3
+
2
1
)
3
1
=
4
9
1
=
9
4
=0.44