What we must first find is the probability of the contestant choosing a key that does not start a car.
We know that there are 10 keys and that 3 of the keys start a car. This means that the number of keys that don't start a car is 10 - 3 = 7 keys.
Now, the probability of her choosing a key that does not start a car is equal to the number of keys that do not start a car divided by the total number of keys. Let us call P(S') the probability of choosing a key that does not start the car; then:
P(S') = 7/10
Remember, the question asks us the probability of a no start, no start outcome, ie. choosing a key that does not start a car, then retuning that key to the bag, mixing it, and then choosing another key randomly that does not start a car.
Since the contestant returns the key to the bag, the probability of choosing a key that does not start a car remains the same. Thus, to find the probability of a no start, no start outcome, we must multiply the probability of choosing a key that does not start a car by the probability of choosing a key that does not start a car, ie. P(S')*P(S'). Thus:
Probability of a no start, no start outcome
= (7/10) * (7/10)
= 49/100
The question asks for the answer as a percentage, therefor we must convert 49/100 to a percentage:
(49/100)*100% = 49%
Thus, the probability of a no start, no start outcome is 49%.
Hope that helped but if you need any more clarification please don't hesitate to comment below.
