Sal has a bag of hard candies: 3 are lemon and 2 are grape. He ate 2 of the candies while waiting for the bus, selecting them at random one after the another.
What is the probability of at least one candy was lemon?
Use complementarity.
Calculate the probability of getting no lemon, and subtract from 1.
Two-step experiment, second step dependent on the first:
Total number of candies at start=5
Step 1:
Number of grapes=2
Probability(No lemon)= 2/5
Step 2:
Number of candies left=4
Number of grapes left=1
Probability(no lemon)=1/4
Probability of both happening=2/5*1/4=1/10
Therefore probability of having at least one lemon = 1-1/10=9/10
I don't understand it says at least one lemon, how can there be none
There are only 3 possibilites:
- no lemon
- 1 lemon
- 2 lemon
so you want the prob (1 lemon OR 2 lemon)
= 2(3/5)(2/4) = 3/5 + (3/5)(2/4) = 3/10
= 9/10
This involved finding the prob of two cases, plus an addition.
You know that the prob of all 3 cases is 1
so what MathMate did was calculate the case you don't want and subtract it from 1 to also get 9/10
that involved the calculation of only one prob and a subtraction.
In this case it didn't make much difference in the length of the solution, but suppose you had 5 different cases and you wanted "at least 1 of those"
Then you would to find prob(1) + prob(2) + ... Prob (5) whereas with the "back-door" or complimentary approach you would have to find only the prob (none) and subtract that from 1.
Thank you
To find the probability of at least one candy being lemon, we can calculate the complementary probability, which is the probability of no lemon candies.
Since Sal ate 2 candies and there are a total of 5 candies, there are 3 candies left in the bag.
The probability of the first candy not being a lemon is 3/5 because there are 3 non-lemon candies left out of the total 5 candies.
The probability of the second candy not being a lemon depends on the first candy being non-lemon. So, if the first candy was not a lemon, there would be 4 candies left with 2 lemon candies remaining. Therefore, the probability of the second candy not being a lemon is 2/4.
To find the probability of both these events happening, we multiply the probabilities together:
P(no lemon candies) = (3/5) * (2/4) = 6/20 = 3/10
Finally, to find the probability of at least one candy being lemon, we subtract the complementary probability from 1:
P(at least one lemon candy) = 1 - P(no lemon candies) = 1 - 3/10 = 7/10
So, the probability of at least one candy being lemon is 7/10.