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Multiple choice business maths probability - iii baye's theorem bayes theorem probability and probability distribution

A signal which can be green or red with probability $\displaystyle \frac{4}{5}$ and $\displaystyle \frac{1}{5}$, respectively, is received at station A and then transmitted to station B. The probability of each station receiving the signal correctly is $\displaystyle \frac{3}{4}$. If the signal received at station B is green, then the probability that the original signal was green is

  1. $\displaystyle \frac{3}{5}$
  2. $\displaystyle \frac{6}{7}$
  3. $\displaystyle \frac{20}{23}$
  4. $\displaystyle \frac{9}{20}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
 Event $G$ = original signal is green
$E _1=A$ receives the signal correct
$E _2=B$ receives the signal correct
E = signal received by B is green
$P(\text{signal received by B is green}) = P(GE _1E _2)+ P(G\cap {E _1}\cap {E _2})+ P(\cap GE _1\cap{E _2})+ P(\cap G\cap {E _1}E _2)$
$P(E)=\dfrac {46}{5\times 16}$
$ P(G/E)=\dfrac {\dfrac {40}5\times 16}{\dfrac {46}5\times16}=\dfrac {20}{23}.$
Multiple choice business maths probability - iii baye's theorem bayes theorem probability and probability distribution

One bag contains 3 white balls, 7 red balls and 15 black balls. Another bag contains 10 white balls, 6 red balls and 9 black balls. One ball is taken from each bag. What is the probability that both the balls will be of the same colour?

  1. $207/625$
  2. $191/625$
  3. $23/625$
  4. $227/625$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Bag $I=$ $3$ White $+$ ${7}$ Red $+$ $15$ Black

Bag $II=$ $10$ White $+$ ${6}$ Red $+$ $9$ Black

Each beg contains total of $25$ balls.

There are three cases for selection of a particular ball :

$1.$ White ball from Bag $I$ and Bag $II=\dfrac{3}{25}\times\dfrac{10}{25}$

$2.$ Red ball from Bag $I$ and Bag $II=\dfrac{7}{25}\times\dfrac{6}{25}$

$3.$ Black ball from Bag $I$ and Bag $II=\dfrac{15}{25}\times\dfrac{9}{25}$

$\therefore$ Total probability $=\dfrac{30}{625}+\dfrac{42}{625}+\dfrac{135}{25}=\dfrac{207}{625}.$

Hence, the answer is $\dfrac{207}{625}.$