Susan has a bag of sweets containing 7 chocolates and 5 toffees. Ahmad has a bag of sweets containing 3 chocolates, 4 toffees and 2 boiled sweets. A sweet is taken at random from Susan’s bag and put in Ahmad’s bag. A sweet is then taken at random from Ahmad’s bag.
(a) Find the probability that the two sweets taken are a toffee from Susan’s bag and a boiled sweet from Ahmad’s bag.
(b) Given that the sweet taken from Ahmad’s bag is a chocolate, find the probability that the sweet taken from Susan’s bag was also a chocolate.
To find the probabilities in both parts of the question, we need to first calculate the total number of possible outcomes and then determine the favorable outcomes.
(a) To calculate the probability of getting a toffee from Susan's bag and a boiled sweet from Ahmad's bag, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
In Susan's bag, there are a total of 7 chocolates and 5 toffees. So, the total number of sweets in Susan's bag is 7 + 5 = 12.
In Ahmad's bag, there are a total of 3 chocolates, 4 toffees, and 2 boiled sweets. So, the total number of sweets in Ahmad's bag is 3 + 4 + 2 = 9.
Since the sweet taken from Susan's bag is put into Ahmad's bag, the total number of possible outcomes is 12 (because there are 12 sweets in Susan's bag) multiplied by 10 (since there are now 10 sweets in Ahmad's bag after adding one sweet from Susan's bag) which equals 120.
Number of favorable outcomes:
To get a toffee from Susan's bag, we have 5 toffees (favorable outcomes).
To get a boiled sweet from Ahmad's bag, we have 2 boiled sweets (favorable outcomes).
So, the total number of favorable outcomes is 5 (number of toffees in Susan's bag) multiplied by 2 (number of boiled sweets in Ahmad's bag) which equals 10.
Probability:
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Probability = Number of favorable outcomes / Total number of possible outcomes
Therefore, the probability of getting a toffee from Susan's bag and a boiled sweet from Ahmad's bag is 10/120, which simplifies to 1/12.
(b) Given that the sweet taken from Ahmad's bag is a chocolate, we need to find the probability that the sweet taken from Susan's bag was also a chocolate.
Total number of possible outcomes:
The total number of possible outcomes remains the same, which is 120 (calculated in part a).
Number of favorable outcomes:
Now, if the sweet taken from Ahmad's bag is a chocolate, there are still 3 chocolates left in Ahmad's bag (favorable outcomes). Additionally, there are still 7 chocolates left in Susan's bag.
So, the total number of favorable outcomes is 3 (number of chocolates in Ahmad's bag) multiplied by 7 (number of chocolates in Susan's bag) which equals 21.
Probability:
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Therefore, the probability of the sweet taken from Ahmad's bag being a chocolate and the sweet taken from Susan's bag also being a chocolate is 21/120, which simplifies to 7/40.
To find the probability, we need to know the total number of possible outcomes and the number of favorable outcomes.
(a) There are 12 sweets in Susan's bag (7 chocolates + 5 toffees) and 9 sweets in Ahmad's bag (3 chocolates + 4 toffees + 2 boiled sweets). The total number of possible outcomes is the product of these two numbers, which is 108 (12 * 9).
There is 1 toffee in Susan's bag and 2 boiled sweets in Ahmad's bag. Therefore, the number of favorable outcomes is the product of these two numbers, which is 2 (1 * 2).
The probability that the two sweets taken are a toffee from Susan’s bag and a boiled sweet from Ahmad’s bag is 2/108, which simplifies to 1/54.
(b) Given that the sweet taken from Ahmad's bag is a chocolate, we can assume that only chocolates from Ahmad's bag were chosen. So, the number of possible outcomes is 3 (the number of chocolates in Ahmad's bag).
The number of favorable outcomes is the number of chocolates in Susan's bag, which is 7.
Therefore, the probability that the sweet taken from Susan's bag was also a chocolate, given that the sweet taken from Ahmad's bag was a chocolate, is 7/3.