MAT prob&stat1 probability

Question 1

What is probability, and how is it described?

Answer 1

Probability deals with the chance or likelihood of an event occurring. It is the language of risk and is used to explain and interpret outcomes in daily life.

Question 2

What does randomness mean in statistics?

Answer 2

An event is random if it happens without conscious decision or design. For example, picking a marble from a bag without seeing ensures randomness if all marbles are identical in size and material.

Question 3

What is theoretical probability, and how is it calculated?

Answer 3

Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of equally likely outcomes:
P(A) = n(A) / n(T).

Question 4

Example: A bag contains 2 purple, 3 green, and 4 blue marbles. What is the probability of picking:
a) A green marble,
b) A yellow marble?

Answer 4

a) P(G) = 3/9 = 1/3.
b) P(Y) = 0/9 = 0 (no yellow marbles in the bag).

Question 5

What are expected frequencies?

Answer 5

Expected frequencies estimate how often an event will occur over a number of trials:
Expected frequency = n × p, where n is the number of trials and p is the event’s probability.

Question 6

Example: A coin is flipped 54 times. What is the expected frequency of heads?

Answer 6

Expected frequency = n × p = 54 × 1/2 = 27 heads.

Question 7

What is experimental probability, and how is it calculated?

Answer 7

Experimental probability is found by dividing the observed frequency of an event by the total number of trials:
P(E) = favorable observations / total observations.

Question 8

Example: A bag of marbles is drawn 1500 times. Observed frequencies are:
Purple: 508, Green: 817, Blue: 175.
Find the experimental probability of drawing a purple marble.

Answer 8

P(Purple) = 508 / 1500 ≈ 0.339.

Question 9

What are mutually exclusive and exhaustive events?

Answer 9

• Mutually exclusive: Two events that cannot occur simultaneously.
  Example: Rolling a die, getting “even” and “odd” are mutually exclusive.
• Exhaustive: Events covering all possible outcomes.
  Example: “Even” and “not even” are exhaustive.

Question 10

What are independent events, and how is their probability calculated?

Answer 10

Independent events are unaffected by the occurrence of one another. For events A and B:
P(A ∩ B) = P(A) × P(B).

Question 11

Example: A coin is flipped and a die is rolled. Find the probability of getting heads and rolling a 6.

Answer 11

P(H ∩ 6) = P(H) × P(6) = (1/2) × (1/6) = 1/12.

Question 12

What notations are used to show combined probabilities of events?

Answer 12

• “And” (A ∩ B): Intersection of two events.
• “Or” (A ∪ B): Union of two events.
• Complement (A’): All outcomes not in A.

Question 13

Example: In a group of 50 students, 30 study math, 25 study English, and 10 study both. Find the probability that a student studies math or English.

Answer 13

n(Math ∪ English) = n(Math ∩ English) + n(Math ∩ Not-English) + n(Not-Math ∩ English)
= 10 + 20 + 15 = 45
P(Math ∪ English) = 45 / 50 = 0.9

Question 14

What is the role of tree diagrams in probability?

Answer 14

Tree diagrams organize and calculate probabilities of sequential events, showing outcomes and their associated probabilities.

Question 15

Example: A bag contains marbles. P(Blue) = 7/10, P(Blue ∩ Swirled) = 3/5, P(Swirled ∩ Not-Blue) = 1/5. Find P(Not-Blue ∩ Not-Swirled).

Answer 15

1. P(Not-Blue ∩ Swirled) = 1/5.
2. P(Not-Blue) = 3/10.
3. P(Not-Blue ∩ Not-Swirled) = P(Not-Blue) - P(Not-Blue ∩ Swirled) = 3/10 - 1/5 = 1/10.

Question 16

What is conditional probability?

Answer 16

Conditional probability is the probability of an event A occurring given that another event B has already occurred. It is denoted as P(A | B) and calculated using:
P(A | B) = P(A ∩ B) / P(B)

Question 17

How is conditional probability represented in tree diagrams?

Answer 17

In a tree diagram:
- The first set of branches shows probabilities for event B.
- The second set shows probabilities for event A given B.
The probability of A ∩ B is found by multiplying probabilities along the branches.

Question 18

Example: A bag contains 50 marbles. There are 30 blue (B), 25 swirled (S), and 10 blue-swirled marbles. Find:
a) P(S | B)
b) P(B | S)

Answer 18

a) P(S | B) = P(B ∩ S) / P(B) = 10/50 / 30/50 = 10/30 = 1/3
b) P(B | S) = P(B ∩ S) / P(S) = 10/50 / 25/50 = 10/25 = 2/5

Question 19

What is the formula for conditional probability in terms of intersections?

Answer 19

P(A ∩ B) = P(A | B) × P(B)

Question 20

Example: In a litter of kittens, there are 7 tabbies and 3 black kittens. Two kittens are chosen randomly. Find the probability that the first kitten is black, given that the second is a tabby.

Answer 20

1. Total outcomes: C(10, 2) = 45
2. P(First is black ∩ Second is tabby) = (3/10) × (7/9) = 21/90 = 7/30
3. P(Second is tabby) = 63/90 = 7/10
4. P(Black | Tabby) = (7/30) / (7/10) = 1/3

Question 21

What are independent events in probability?

Answer 21

Two events are independent if the occurrence of one does not affect the probability of the other:
P(A | B) = P(A), or equivalently, P(A ∩ B) = P(A) × P(B).

Question 22

Example: In a class of 50 students, 30 study maths, 15 study French, and 9 study both. Determine if studying maths (M) and French (F) are independent.

Answer 22

1. P(M) = 30/50 = 3/5.
2. P(F) = 15/50 = 3/10.
3. P(M ∩ F) = 9/50.
4. Check independence: P(M ∩ F) = P(M) × P(F)?
   (9/50) = (3/5 × 3/10) → Events are independent.

Question 23

How are conditional probabilities tested for independence?

Answer 23

Events A and B are independent if:
P(A | B) = P(A), or equivalently, P(B | A) = P(B).

Question 24

Example: A teacher has analysed some results of students taking AS Maths and A level Maths, collecting information on students who have achieved a grade B or higher (B+). She has estimated the following probabilities.
* The probability that a student gets B+ on AS Maths is 0.6.
* If the student has achieved B+ at AS Maths, the probability that the student gets a B+
grade at A level Maths is 0.7.
* If the student has not achieved a grade B+ at AS, the probability that a student gets a
B+ grade at A level is 0.35.
Calculate the probability that an B+ grade is achieved at AS, given that an B+ grade has been achieved at A level.

Answer 24

• P(B+ at AS) = 0.6,
• P(B+ at A-level | B+ at AS) = 0.7,
• P(B+ at A-level | Not B+ at AS) = 0.35.
  Find P(B+ at AS | B+ at A-level):
  1. P(AS ∩ A-level) = P(AS) × P(A-level | AS) = 0.6 × 0.7 = 0.42
  2. P(A-level) = P(AS ∩ A-level) + P(Not AS ∩ A-level)
  = 0.42 + (0.4 × 0.35) = 0.56
  3. P(AS | A-level) = P(AS ∩ A-level) / P(A-level)
  = 0.42 / 0.56 = 0.75

Question 25

What is the union formula in probability?

Answer 25

For any events A and B:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Question 26

Example: A sports center interviews 25 people. Of the 11 children, 7 prefer team sports. 8 adults prefer individual workouts. Given someone prefers individual workouts, what is the probability they are an adult?

Answer 26

1. Total individual workout preferences = 12.
2. P(Adult | Individual) = P(Adult ∩ Individual) / P(Individual).
   = 8/12 = 2/3.