Q.3.24 - A First Course in Probability

Question

 Rank the following from most likely to least likely to occur:

1. A fair coin lands on heads.

2. Three independent trials, each of which is a success with probability$.8$, all result in successes.

3. Seven independent trials, each of which is a success with probability $9$, all result in successes.

Step-by-Step Solution

Answer

Three independent trails are ranked first, second, and third$0.4783<0.5<0.512$ $P3<P1<P2$ Use the definition of independence in conjunction with the probability of intersections

Step1:A fair coin lands on heads. (part a)

$P_1=P($fair coin lands on head=$1/2$

Step2:Three independent trails(part b)

independence

$E_1,E_2,\dots ,E_n$are independent$\iff P\left(E_1{E}_2\dots E_n\right)=\prod _{i=1}^{n}P\left(E_i\right)$

P(one trial ends in success )=$0.8$

Therefore

$P_2=P(3$Independent trials are successful)=P One trial is successful.  $^3=0.8^3=0.512$

Step3:Seven independent trail(part c)

$E_1,E_2,\dots ,E_n\text{ are independent }\iff P\left(E_1{E}_2\dots E_n\right)=\prod _{i=1}^n P\left(E_i\right)$

Given

P( one trial ends in success )=$0.9$

Therefore

$P_3=P(7$Independent trials are successful. One trial is successful.)$^7$$={0.9}^7\approx 0.4783$

$\begin{array}{c}0.4783<0.5<0.512\\ P_3<P_1<P_2\end{array}$