Biased Coin Probability: Exactly Two Tails

Let's break down the probability calculation for tossing a biased coin three times and getting exactly two tails.

Understanding the Biased Coin

We are given that a head (H) is twice as likely to occur as a tail (T). This can be written as:

\( P(H) = 2 \times P(T) \)

Also, the sum of probabilities for all possible outcomes must be 1:

\( P(H) + P(T) = 1 \)

Now, we can substitute the first equation into the second:

\( 2 \times P(T) + P(T) = 1 \)

\( 3 \times P(T) = 1 \)

So, the probability of getting a tail is:

\( P(T) = \frac{1}{3} \)

And the probability of getting a head is:

\( P(H) = 2 \times \frac{1}{3} = \frac{2}{3} \)

Identifying Outcomes with Exactly Two Tails

When a coin is tossed three times, the possible sequences for getting exactly two tails are:

• Tail, Tail, Head (TTH)
• Tail, Head, Tail (THT)
• Head, Tail, Tail (HTT)

Calculating Probability for Each Outcome

Since each toss is independent, we can multiply the probabilities of the individual outcomes for each sequence:

• For TTH: \( P(\text{TTH}) = P(T) \times P(T) \times P(H) = \frac{1}{3} \times \frac{1}{3} \times \frac{2}{3} = \frac{2}{27} \)
• For THT: \( P(\text{THT}) = P(T) \times P(H) \times P(T) = \frac{1}{3} \times \frac{2}{3} \times \frac{1}{3} = \frac{2}{27} \)
• For HTT: \( P(\text{HTT}) = P(H) \times P(T) \times P(T) = \frac{2}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{2}{27} \)

Finding the Total Probability

To find the probability of getting exactly two tails, we sum the probabilities of these three mutually exclusive outcomes:

\( P(\text{exactly two tails}) = P(\text{TTH}) + P(\text{THT}) + P(\text{HTT}) \)

\( P(\text{exactly two tails}) = \frac{2}{27} + \frac{2}{27} + \frac{2}{27} \)

\( P(\text{exactly two tails}) = \frac{2 + 2 + 2}{27} = \frac{6}{27} \)

Simplifying the Result

The fraction \(\frac{6}{27}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

\( \frac{6 \div 3}{27 \div 3} = \frac{2}{9} \)

Thus, the probability of getting exactly two tails when the biased coin is tossed three times is \(\frac{2}{9}\).