5.26 - A First Course in Probability

Question

Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads $55$ percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin $1000$ times. If the coin lands on heads $525$ or more times, then we shall conclude that it is a biased coin, whereas if it lands on heads fewer than $525$ times, then we shall conclude that it is a fair coin.

Step-by-Step Solution

Answer

If the coin is actually fair, then the probability that we shall reach a false conclusion is $0.0606$. If the coin were biased, then the probability that we shall reach a false conclusion is $0.0525$.

Step 1. Given Information.

Here, it is given that - 

Two types of coins are produced at a factory: fair and biased.

Each coin comes up heads $55\%$ of the time.

Number of tosses = $1000$

The coin is biased, if it lands on heads for more than $525$ times.

The coin is fair, if it lands on heads for less than $525$ times.

Step 2. Calculate mean and standard deviation of the fair coin.

The probability of getting head in a fair coin is $p=0.5$

Let $A$ be the event that the test concludes the coin is fair.

Let $X_A$ be the number of heads in the test dependent on fair coin.

Number of times the coin is tossed = $n$ = 1000

$$\begin{gathered} \mu =np \\ {\mu }_A=1000\times 0.5=500 \end{gathered}$$

$$\begin{gathered} \sigma =\sqrt{np(1-p)} \\ {\sigma }_A=\sqrt{(1000\times 0.5)(1-0.5)} \\ {\sigma }_A=\sqrt{250} \\ {\sigma }_A=15.8114 \end{gathered}$$

Step 3. Calculate the probability that the researcher reaches a false conclusion when the coin is fair.

$$\begin{gathered} P\left(X_A\right)=P(X_A\ge 525-0.5) \\ =P(X_A\ge 524.5) \end{gathered}$$

$$\begin{gathered} P(X_A\ge 524.5)=P\left(\frac{X_A-{\mu }_A}{{\sigma }_A}\right) \\ =P\left(\frac{524.5-500}{15.8114}\right) \\ =P(Z\ge 1.5495) \\ =1-\phi (1.5495) \\ =0.0606 \end{gathered}$$

 Therefore, the probability that we reach a false conclusion when the coin is fair is $0.0606$.

Step 4. Calculate mean and standard deviation of the biased coin.

The probability of getting head in a biased coin is $=0.55$

Let $B$ be the event that the test concludes the coin is biased.

Let $X_B$ be the number of heads in the test dependent on biased coin.

Number of times the coin is tossed = $n = 1000$

$$\begin{gathered} \mu =np \\ {\mu }_B=1000\times 0.55=550 \end{gathered}$$

$$\begin{gathered} \sigma =\sqrt{np(1-p)} \\ {\sigma }_B=\sqrt{(1000\times 0.55)(1-0.55)} \\ {\sigma }_B=\sqrt{247.5} \\ {\sigma }_B=15.7321 \end{gathered}$$

Step 5. Calculate the probability that the researcher reaches a false conclusion when the coin is biased.

$$\begin{gathered} P\left(X_B\right)=P(X_B<525-0.5) \\ =P(X_B<524.5) \end{gathered}$$

$$\begin{gathered} P(X_B<524.5)=P\left(\frac{X_B-{\mu }_B}{{\sigma }_B}\right) \\ =P\left(\frac{524.5-550}{15.7321}\right) \\ =P(Z<-1.6209) \\ =0.0525 \end{gathered}$$

Therefore, the probability that we reach a false conclusion when the coin is biased is $0.0525$.