CAT Mathematical Reasoning: Combinations and Probability (Chance)
Combinations
The Combinations questions asked are usually pretty simple. One question of this type may be asked. They can generally be answered by simple multiplication.
Q1: A furniture shop sells three types of dining tables and four types of chairs. If Jim wants to buy one dining table, and some chairs of one type, in how many ways can he make the purchase?
Answer:
The number of ways Jim can select a dining table = 3 (because there are 3 types of tables)
The number of ways Jim can select a chair = 4 (there are 4 types of chairs)
Hence, the total number of ways in which Jim can select the dining table + chair combo
= 4 * 3 = 12
To explain the above in a different way:
Let the dining tables be: D1, D2, D3
Let the chairs be: C1, C2, C3, C4
If D1 is selected, then any of C1, C2, C3, C4 can be selected: this is 4 ways
If D2 is selected, then any of C1, C2, C3, C4 cab be selected: this is another 4 ways
Same for D3.
Hence the total number of ways = 4 + 4 + 4 = 12
Try this on your own:
Q-1: A restaurant offers diners a choice of three entrees, two soups, four main courses and two desserts. In how many ways can a diner eat, if he has a complete meal?
Q-2: A child has to choose three toys in the following way: one marble out of five differently colored marbles, one stuffed toy out of three different stuffed toys and one ball out of four differently shaped balls. In how many ways can he choose the three toys?
Q-3: A senior in college needs to select four optional subjects for study in her final semester. She can choose from four subjects offered by the Mathematics department, two by the Physics department, three by the Chemistry department and two by the Humanities department, but she has to choose exactly one subject from each department. In how many ways can she make her selection?
CCAT Probability:
Probability (also referred to as Chance) is a measure of the likelihood of something happening.
Formally, probability of an event =
the number of ways in which this event can occur / total number of ways in which all events can occur
Probability is usually expressed as a number between 0 and 1 (both inclusive).
0 = there is no chance of the event occurring
1 = the event will definitely occur
Now, probability of an event occurring + probability of the event not occurring = 1
Probability can also be expressed as a percentage. (In this case, probability of an event occurring + probability of the event not occurring = 100%)
A probability of 0.2 = 0.2 * 100 = 20%
A probability of 30% = 30/100 = 0.3
Q1: A coin is tossed. What is the probability that it comes up Heads?
Answer:
The total number of events which can occur if a coin is tossed = 2 : it can come up Heads, or, it can come up Tails.
Of this, the number of ways in which the desired event (the coin comes up Heads) can occur = 1
Hence, the required Probability = 1 / 2 or 0.5 or 50%
Q2: A six sided dice is rolled. What is the probability that a 4 is rolled?
Answer:
The total number of events = 6 (any of 1, 2, 3, 4, 5, 6 can be rolled)
The number of ways in which 4 is rolled = 1
Hence, the Probability = 1 / 6
Q3: Two dice are rolled. What is the probability that their sum is 5?
Answer:
The total number of events here = 6 * 6 = 36 (each dice can roll between 1 and 6)
A sum of 5 can be obtained in the following ways:
1+4 : 2+3 : 3+2 : 4+1
Thus, a total of 4 ways.
Hence the required probability = 4 / 36 = 1/9
Sometimes, slightly more complicated questions are asked, involving two independent events occurring at the same time.
Independent means that the occurrence of one event does not depend on or influenced by the other.
For example:
A couple has two children. The first child could be a boy or a girl. Independent of this, the second child could be a boy or girl. So, the two children being boys or girls are independent events.
A dice is rolled two times in succession. The two rolls are independent, because neither roll depends on the other.
In the case of independent events, the probability of two events happening= product of individual probabilities
Q4: A couple has two children 5 years apart. What is the probability that both are girls?
Answer:
Method 1:
The probability that the first child is a girl = 1 / 2
The probability that the second child is a girl = 1 / 2
Since the events are independent, the probability that both are girls = (1 / 2) * (1 / 2) = 1 / 4
Method 2:
Let the birth of a girl and that of a boy be represented by G and B respectively.
Then the two births could happen in the following ways:
BG (i.e. the first child is a Boy, the second a Girl)
BB
GB
GG
So, the total number of possible events = 4
The number of favorable events = 1 (the one in bold)
So, the probability for this = 1 / 4
Q5: A dice is rolled twice in succession. What is the probability that a 6 is rolled both times?
Answer:
The probability that a 6 is rolled on either roll = 1 / 6
The two rolls are independent.
Hence, the probability that a 6 is rolled both times = 1 / 6 * 1 / 6
= 1 / 36
Q6: There is a 20% chance that a person is left handed. There is a 25% chance that a person has blue eyes. What is the chance that a person selected at random is left handed and has blue eyes?
Answer:
The chance that the person is left handed = 20% = 20/100 = 0.2
The chance that the person has blue eyes = 25% = 25/100 = 0.25
The two events are independent.
Hence the probability that the person is both left handed and has blue eyes = 0.2 * 0.25 = 0.050 = 0.05 * 100 = 5%
Q7: An estimated 30% of all people in a particular town consider football to be their favorite sport. If two people are selected at random from this town, what are the chances that NEITHER of these people consider football to be their favorite sport?
Answer:
Note the word “NEITHER” in the question. We need to find the chance (probability) that neither of the selected people consider football to be their favorite sport.
The probability that football is NOT the favorite sport of a randomly selected person = 100–30 = 70% = 0.7
Hence, if two people are selected at random, the probability that football is not the favorite sport of either = 0.7 * 0.7 = 0.49 = 49%
Now, try these:
Q-1: On his birthday, Mark distributes one sweet each to his 60 classmates. He distributes 15 Snickers, 15 Mars Bars, 20 Hersheys and 10 Cadburys at random. His best friend in the class is Zack. What is the probability that Zack gets a Hershey’s chocolate?
Q-2: A basket contains 10 red, 20 blue, 30 green and 40 violet coloured toys. A child takes out a toy at random. What is the probability that the selected toy is not green?
Q-3: It is estimated that there is a 10% chance that any TV program will be delayed. If a person watches two TV programs, what are the chances that BOTH TV programs will be delayed?
Solutions to Combinations Problems:
Q-1: A restaurant offers diners a choice of three entrees, two soups, four main courses and two desserts. In how many ways can a diner eat, if he has a complete meal?
Answer-1:
The diner can eat in 3 * 2 * 4 * 2 = 48 ways
Q-2: A child has to choose three toys in the following way: one marble out of five differently colored marbles, one stuffed toy out of three different stuffed toys and one ball out of four differently shaped balls. In how many ways can he choose the three toys?
Answer-2:
The child can choose in 5 * 3 * 4 = 60 ways
Q-3: A senior in college needs to select four optional subjects for study in her final semester. She can choose from four subjects offered by the Mathematics department, two by the Physics department, three by the Chemistry department and two by the Humanities department, but she has to choose exactly one subject from each department. In how many ways can she make her selection?
Answer-3:
The student can choose in 4 * 2 * 3 * 2 = 48 ways
Solutions to Probability Problems:
Q-1: On his birthday, Mark distributes one sweet each to his 60 classmates. He distributes 15 Snickers, 15 Mars Bars, 20 Hersheys and 10 Cadburys at random. His best friend in the class is Zack. What is the probability that Zack gets a Hershey’s chocolate?
Answer-1:
The probability here
= Number of Hershey sweets / Total number of sweets
= 20 / 60
= 1 / 3
Q-2: A basket contains 10 red, 20 blue, 30 green and 40 violet coloured toys. A child takes out a toy at random. What is the probability that the selected toy is not green?
Answer-2:
The probability
= Number of non-green toys / Total number of toys
= (10 + 20 + 40) / (10 + 20 + 40 + 30)
= (70 / 70 + 30)
= 70 / 100
= 70% or 0.7
Q-3: It is estimated that there is a 10% chance that any TV program will be delayed. If a person watches two TV programs, what are the chances that BOTH TV programs will be delayed?
Answer-3:
The probability that a TV program will be delayed = 10% = 0.1
The two programs being delayed are independent events
Hence the probability that bot programs will be delayed =
0.1 * 0.1 = 0.01 = 1%
Coming up next: CCAT Mathematical Reasoning Weighted Averages
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