Question

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

Short answer

The probability is$P\left(\mathrm{T}\right)=\frac{1}{5},P\left(\mathrm{H}\right)=\frac{4}{5}$.

Step-by-step solution

Step 1. Given information.

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

Step 2. Find probability.

We are given that the heads H is four times as likely as tailsTto occur in a coin. Thus, we can let

$$\begin{gathered} P\left(\mathrm{T}\right)=x \\ P\left(\mathrm{H}\right)=4x \end{gathered}$$

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 $P\left(\mathrm{T}\right)\mathrm{and}P\left(\mathrm{H}\right)$and solving for $x$, we have

$$\begin{gathered} P\left(\mathrm{T}\right)+P\left(\mathrm{H}\right)=1 \\ x+4x=1 \\ 5x=1 \\ x=\frac{1}{5} \end{gathered}$$

Since $x=\frac{1}{5}$, assign the probabilities

localid="1647248180167" $P\left(\mathrm{T}\right)=x=\frac{1}{5},P\left(\mathrm{H}\right)=4x=4\left(\frac{1}{5}\right)=\frac{4}{5}$.

Step 3. Conclusion.

The probability is$P\left(\mathrm{T}\right)=\frac{1}{5},P\left(\mathrm{H}\right)=\frac{4}{5}$.