Question

Two coins R and S are tossed. The 4 joint events $H_R{H}_S,T_R{T}_S,H_R{T}_S,T_R{H}_S$ have probabilities 0.28, 0.18, 0.30, 0.24 respectively, where H represents head and T represents tail. Which one of the following is TRUE?

• A. The coin tosses are independent
• B. R is fair, S is not
• C. S is fair, R is not
• D. The coin tosses are dependent

Answer 1

The correct option is D The coin tosses are dependent

$P\left(H_R{H}_S\right)=0.28$
$P\left(T_R{T}_S\right)=0.18$
$P\left(H_R{T}_S\right)=0.30$
$P\left(T_R{H}_S\right)=0.24$

Two events A and B are said to be independent if
$P(A\cap B)=P\left(A\right)P\left(B\right)$

$P\left(H_R\right)=P\left(H_S\right)=P\left(T_R\right)=P\left(T_S\right)=\frac{1}{2}$

So option (a) is wrong.
Now, if R is fair and S is not then

$P\left(H_R\right)=P\left(T_R\right)=\frac{1}{2}$

$P\left(H_R{H}_S\right)=\frac{1}{2}P\left(H_S\right)=0.28$

$P\left(H_S\right)=0.56$

$P\left(H_R{T}_S\right)=\frac{1}{2}P\left(T_S\right)=0.30$

$P\left(T_S\right)=0.60$

$P\left(H_S\right)+P\left(T_S\right)=1.61>1$

which is not possible.

Option (b) is wrong

Similarly option (c) is also wrong.