Finding the probability that X will be successful if its success is predicted Consider an electronics company is planning to introduce a new
camera phone. The company commissions a marketing
report for each newproduct that predicts either the success
or the failure of the product. Of new products introduced
by the company, 60% have been successes. Furthermore,
70% of their successful products were predicted to be
successes, while 40% of failed products were predicted
to be successes. Find the probability that this new camera
phone will be successful if its success has been predicted.
To solve this, my approach would be to:
x = total no. of products
0.6x = successful products 
0.4x = failed products
so total products which are successful as predicted are 0.6x*70/100 or 0.42x
and total products whose success was predicted but they have failed are 0.4x*40/100 or 0.16x
therefore total products whose success was predicted are
0.42x + 0.16x or 0.58 x
so the percentage of successful products out of those whose success had been predicted is 0.42x/0.58x * 100 or 72.414‫‌‍%
so the probability of a product being success as predicted is 0.72414
Is this right?  Thank you!
 A: Let $M$ be the event the new product is predicted to be successful and $S$ be the event that the new product is actually successful. 
We gather from the question that $P(S)=0.6$, $P(M|S)=0.7$ and $P(M|S')=0.4$ We need to find $P(S|M)$
$$\begin{align*}
P(M|S)\times P(S)&= P(M\cap S) \\\\ P(M\cap S)&=0.42 =  P(S\cap M)\\\\P(M) &= P(S\cap M) + P(S'  \cap M)\\&=0.42 + (0.4)(0.4)\\&=0.58 \\\\ \therefore P(S|M)&=\frac{P(S \cap M)}{P(M)}\\&=\frac{0.42}{0.58}\\&=0.724 \text{ or }72.4\%
\end{align*}$$
Which agrees with your answer.
A: <>   Let total new products introduced by the company are 1000.
<>   now total number of products which are succeful are 60% of total i.e 600.
<>   now total number of products which are failed  are  40% of total i.e 400.
<>   succeful products whose success are predicted are 70% of total successful products i.e 600*0.7 = 420
<>   failed products whose success are predicted are 40% of total unsuccessful products i.e 400*0.4 = 160
<>   (sample space for this problem ) [ S ] = total number of products whose success is predicted = 160 + 420 = 580
<>   (Our event)[E] = total number of product whose success is predicted and are successful = 420
p(E) = E / S = 420 / 580 = 0.7241
enter image description here
