Question 1196465
1. An aerospace company has submitted bids on two separate federal government defense contracts. 
The company president believes that there is a 57% probability of winning the first contract. 
If they win the first contract, the probability of winning the second is 67%. 
However, if they lose the first contract, the president thinks that the probability 
of winning the second contract decreases to 44%.
What is the probability that they lose both contracts?
~~~~~~~~~~~~~~~~~



            When I read the solution by @math_tutor2020, everything looks good,
            but then I read the last line, where he multiplies the individual loosing probabilities,
            as if these events are independent - which is not obvious for me.


            Therefore, I developed another solution. It produces the same answer,
            but, at least, withdraws the question about independency of loosing events.



<pre>
P(win 1st contract)  = P(1) = 0.57  (given).

P(lose 1st contract) = 1 - 0.57 = 0.43  (the complement).


P(win  2nd contract) = P(2) = 0.57*0.67 + (1-0.57)*0.44 = 0.5711.


P(win both 1st and 2nd contracts) = 0.57*0.67 = 0.3819.


P(win at least one of the two contracts) = P(1) + P(2) - P(both) = 0.57 + 0.5711 - 0.3819 = 0.7592.

P(lose both) = 1 - P(win at least one of the two contracts) = 1 - 0.7592 = 0.2408 = 24.08%.     <U>ANSWER</U>
</pre>

Solved.