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A bag contains one fair coin, two two-headed coins, and three two-tailed coins. Each of the six coins is flipped, but the outcomes of five of the coins are hidden from you, randomly. If the outcome you see is heads, what is the probability that the fair coin (which may or may not be the coin that was shown to you) landed heads up?

My attempt would be there are $12$ total faces and your looking for $5$ of those $12$. Not sure where to go from there

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Using Bayes' Theorem:

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$

we have the following events:

A is the event the fair coin lands heads
B is the event that we see heads

So $P(B|A)=\dfrac12$ ($3$ from $6$) and $P(A)=\dfrac12$.

$P(B)$ can be expanded into:

$P(B)=P(B|A)P(A)+P(B|A^c)(P(A^c)$

with $P(B|A^c)=\dfrac13$ ($2$ from $6$) and $P(A^c)=\dfrac12$.

So $P(B)=\dfrac12\dfrac12+\dfrac13\dfrac12=\dfrac14+\dfrac16=\dfrac5{12}$.

Final answer is:

$$P(A|B)=\frac14\frac{12}5=\frac35$$

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Consider the possible outcomes:

  • There are two heads and four tails ($\frac{1}{2}$ chance)
    • You see heads ($\frac{2}{6}\cdot\frac{1}{2}=\frac{1}{6}$ chance)
    • You see tails ($\frac{4}{6}\cdot\frac{1}{2}=\frac{1}{3}$ chance)
  • There are three heads and three tails ($\frac{1}{2}$ chance)
    • You see heads ($\frac{3}{6}\cdot\frac{1}{2}=\frac{1}{4}$ chance)
    • You see tails ($\frac{3}{6}\cdot\frac{1}{2}=\frac{1}{4}$ chance)

Given that we see heads, the ratio of the chances that the fair coin landing tails to the fair coin landing heads is therefore equal to $\frac{1}{6}:\frac{1}{4}$. Scaling so that the sum is 1, we have $\frac{2}{5}:\frac{3}{5}$.

So there is a 60% chance that the fair coin landed heads.

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My attempt would be there are $12$ total faces and your looking for $5$ of those

There are twelve total faces, of which five are heads, and the fair coin is one of these five.

Given that the coin shown is heads up; What is the probaility that this heads belongs to the fair coin?

Under the same condition; What is the probability that the heads showing does not belong to the fair coin, and yet the fair coin is heads up?

Put the results together.

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