Question

A fair coin is tossed five times. Find the possibilities that a head appears
i) exactly three times
ii) at least two times
iii) at the most four times.

Answer 1

Solution. Let event A be the appearance of a head. The probability of event A is 0.5 for each throw. Since the probability is the same for all throws, we use the formula

$$P_n(m) = C_n^m*p^m*q^{n-m}$$

where n=5 is number of events; m is number of event A; C is number of combinations; p=0.5 is probability of the event A; q=1-p=0.5. Find the probability for all possible quantities of event A with 5 throws. Get

$$P_5(0) = C_5^0*0.5^0*0.5^5=0.03125$$

$$P_5(1) = C_5^1*0.5^1*0.5^4=5*0.03125=0.15625$$

$$P_5(2) = C_5^2*0.5^2*0.5^3=10*0.03125=0.31250$$

$$P_5(3) = C_5^3*0.5^3*0.5^2=10*0.03125=0.31250$$

$$P_5(4) = C_5^4*0.5^4*0.5^1=5*0.03125=0.15625$$

$$P_5(5) = C_5^5*0.5^5*0.5^0=0.03125$$

Therefore

i)

$$P_5(3) =0.31250$$

ii)

$$P_5(2)+P_5(3) +P_5(4)+P_5(5)= 0.8125$$

iii)

$$P_5(0)+P_5(1) +P_5(2)+P_5(3)+P_5(4)=0.96875$$

Answer.  i) 0.3125 ii) 0.8125 iii) 0.96875