91Ó°ÊÓ

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.

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

localid="1646632592661" $$\begin{gathered} \mathrm{μ}=np \\ {\mathrm{μ}}_A=1000×0.5=500 \end{gathered}$$

localid="1646632650748" $$\begin{gathered} \mathrm{σ}=\sqrt{np(1-p)} \\ {\mathrm{σ}}_A=\sqrt{(1000×0.5)(1-0.5)} \\ {\mathrm{σ}}_A=\sqrt{250} \\ {\mathrm{σ}}_A=15.8114 \end{gathered}$$

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

$$\begin{gathered} P(X_Aâ‰\yen 524.5)=P\left(\frac{X_A-{\mathrm{μ}}_A}{{\mathrm{σ}}_A}\right) \\ =P\left(\frac{524.5-500}{15.8114}\right) \\ =P(Zâ‰\yen 1.5495) \\ =1-\mathrm{φ}(1.5495) \\ =0.0606 \end{gathered}$$

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

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 = $n=1000$

$$\begin{gathered} \mathrm{μ}=np \\ {\mathrm{μ}}_B=1000×0.55=550 \end{gathered}$$

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

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

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

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