Question

Assigning Probabilities A coin is weighted so that tails is twice as likely as heads to occur. What probability should we assign to heads? to tails?

Short answer

The probability is $P\left(\mathrm{T}\right)=\frac{2}{3},P\left(\mathrm{H}\right)=\frac{1}{3}$.

Step-by-step solution

Step 1. Given information. 

A coin is weighted so that tails is twice as likely as heads to occur. What probability should we assign to heads? to tails?

Step 2. Find probability. 

We are given that the tail Tis twice times as likely as heads Hto occur in a coin. Thus, we can let

$$\begin{gathered} P\left(\mathrm{H}\right)=x \\ P\left(\mathrm{T}\right)=2x \end{gathered}$$

Since the sum of the probabilities of the possible outcomes must equal 1, we have

$P\left(\mathrm{T}\right)+P\left(\mathrm{H}\right)=1$

Substituting the value of localid="1647255351165" $P\left(\mathrm{T}\right)\mathrm{and}P\left(\mathrm{H}\right)$and solving for x ,we have

$$\begin{gathered} P\left(\mathrm{T}\right)+P\left(\mathrm{H}\right)=1 \\ 2x+x=1 \\ 3x=1 \\ x=\frac{1}{3} \end{gathered}$$

Since $x=\frac{1}{3}$, assign the probabilities

$P\left(\mathrm{T}\right)=2x=\frac{2}{3},P\left(\mathrm{H}\right)=x=\frac{1}{3}$.

Step 3. Conclusion. 

The probability is $P\left(\mathrm{T}\right)=\frac{2}{3},P\left(\mathrm{H}\right)=\frac{1}{3}$.