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⬅ Back to Statistics and Probability Topics

Basic Probability

Probability is a number from 0 to 1 that measures how likely an event is. P = 0 means impossible (rolling a 7 on a standard die). P = 1 means certain (rolling any number from 1–6). P = 0.5 is a 50-50 chance (like a fair coin flip). For equally likely outcomes, count how many outcomes satisfy your event and divide by the total. The complement rule ($P(\text{not }A) = 1 - P(A)$) is often easier — instead of counting what you want, count what you don't want and subtract. The addition rule adds probabilities of two events but subtracts the overlap to avoid double-counting.

$$P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$$
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Single Die Probability

A fair die is rolled. Find the probability of getting an even number or a number greater than 4.

Even numbers: {2,4,6}. Greater than 4: {5,6}. Union: {2,4,5,6}.

$$P=\frac{4}{6}=\frac{2}{3}$$

Cards: Heart or King

One card is drawn from a standard 52-card deck. Find the probability that it is a heart or a king.

$$P(H\cup K)=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}$$

Final answer: $4/13$.

Complement Probability

A box contains 5 defective and 45 good bolts. If one bolt is selected, find the probability that it is not defective.

$$P(\text{not defective})=1-\frac{5}{50}=0.90$$

Final answer: 0.90 or 90%.

Two Dice: Sum of 7 or 11

Two fair dice are rolled. Find the probability the sum is 7 or 11.

Total outcomes = 36. Ways to get sum 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 ways. Ways to get sum 11: (5,6),(6,5) = 2 ways. Events are mutually exclusive (no overlap).

$$P=\frac{6+2}{36}=\frac{8}{36}=\frac{2}{9}$$

Final answer: $2/9 \approx 0.222$.

Mutually Exclusive Events

In a single roll of a die, find the probability of getting a 2 or a 5.

These events are mutually exclusive (both cannot happen at once), so $P(A \cap B) = 0$.

$$P(2\cup5)=P(2)+P(5)=\frac{1}{6}+\frac{1}{6}=\frac{2}{6}=\frac{1}{3}$$

Final answer: $1/3$.

Independent Events

A coin is flipped and a die is rolled. Find the probability of getting heads AND a 3.

The two events are independent (the coin doesn't affect the die). For independent events: $P(A \cap B) = P(A) \cdot P(B)$.

$$P(\text{H and 3})=\frac{1}{2}\times\frac{1}{6}=\frac{1}{12}$$

Final answer: $1/12$.

Probability from a Venn Diagram

In a group of 40 students: 18 like Math, 22 like Science, and 10 like both. Find the probability a randomly chosen student likes at least one subject.

Use the addition rule to find the union.

$$n(M\cup S)=18+22-10=30$$
$$P(M\cup S)=\frac{30}{40}=0.75$$

Final answer: 0.75 or 75%.

Three Coin Flips: At Least Two Heads

A fair coin is flipped 3 times. Find the probability of getting at least 2 heads.

Total outcomes = $2^3 = 8$. Favorable: exactly 2 heads: HHT, HTH, THH (3 ways); exactly 3 heads: HHH (1 way). Total = 4.

$$P(\text{at least 2 H})=\frac{4}{8}=\frac{1}{2}$$

Final answer: $1/2$.

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Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q694

MSTE - Statistics and Probability / Probability / Engr. Janclyde Espinosa (Clidez)

If 23 people are in a room, the chances are better than even that at least two people share the same birthday. How many people must be present in order to provide at least an even chance that two or more were born on the same day of the week?

  1. 4
  2. 14
  3. 22
  4. 6
For days of the week, no shared birthday weekday among $n$ people has probability:
$\frac{7\cdot6\cdot5\cdots(7-n+1)}{7^n}$
For $n=3$, probability of a match is $1-\frac{7\cdot6\cdot5}{7^3}=0.388$. For $n=4$:
$1-\frac{7\cdot6\cdot5\cdot4}{7^4}=0.650$
The first at least even chance is:
$\boxed{4}$

Question Bank: q709

MSTE - Statistics and Probability / Probability / Engr. Janclyde Espinosa (Clidez)

Assume that a single depth charge has a probability 1/2 of sinking a submarine, 1/4 of damaging it, and 1/4 of missing. Assume also that two damaging explosions sink the sub. What is the probability that 4 depth charges will sink the sub?

  1. 251/256
  2. 51/256
  3. 25/256
  4. 125/256
A submarine is not sunk only if fewer than two damaging events occur and no direct sink occurs. For each charge: sink $=1/2$, damage $=1/4$, miss $=1/4$. Not sunk after 4 charges means all non-sink outcomes with 0 or 1 damage:
$P(0D,4M)=(1/4)^4$
$P(1D,3M)=\binom{4}{1}(1/4)(1/4)^3$
$P(\text{not sunk})=5/256$
$P(\text{sunk})=1-5/256$
$\boxed{251/256}$

Question Bank: q714

MSTE - Statistics and Probability / Fundamental Principles of Counting / Engr. Janclyde Espinosa (Clidez)

There are 3 red chips and 2 blue chips. When arranged in a row, they form a certain color pattern, for example RBRRB. How many color patterns are possible?

  1. 10
  2. 12
  3. 24
  4. 60
Arrange 3 red and 2 blue chips. Number of distinct color patterns is:
$\frac{5!}{3!2!}=10$
$\boxed{10}$

Question Bank: q718

MSTE - Statistics and Probability / Probability / Engr. Janclyde Espinosa (Clidez)

Kate and David each have €10. Together they flip a coin 5 times. Every time the coin lands on heads, Kate gives David €1. Every time the coin lands on tails, David gives Kate €1. After the coin is flipped 5 times, what is the probability that Kate has more than €10 but less than €15?

  1. 15/32
  2. 5/16
  3. 1/2
  4. 21/32

Solution pending in psadquestions/q718.json.

Question Bank: q721

MSTE - Statistics and Probability / Fundamental Principles of Counting / Engr. Janclyde Espinosa (Clidez)

Paula has 3 movies that she can watch during the weekend: 1 Action, 1 Comedy, 1 Drama. However, she needs to watch the Drama 3 times. Assuming Paula has time for 5 movies and intends to watch all of them, in how many ways can she do so?

  1. 20
  2. 6
  3. 24
  4. 60
The five movie slots consist of A, C, and three D's. Distinct arrangements:
$\frac{5!}{3!}=20$
$\boxed{20}$

Question Bank: q725

MSTE - Statistics and Probability / Statistics / Engr. Janclyde Espinosa (Clidez)

A spring of natural length 10 cm is such that a force of 6 kN stretches it 2 cm. Find the work (in N·m) necessary to stretch the spring from a length of 14 cm to 18 cm.

  1. 720
  2. 680
  3. 240
  4. 350
Natural length is 10 cm. A force of 6000 N stretches it 2 cm = 0.02 m, so:
$k=6000/0.02=300000\text{ N/m}$
Stretching from 14 cm to 18 cm means extension from 0.04 m to 0.08 m. Work:
$W=\frac{1}{2}k(x_2^2-x_1^2)$
$W=\frac{1}{2}(300000)(0.08^2-0.04^2)$
$\boxed{720}$

Question Bank: w17

MSTE - Statistics and Probability / Probability / MSTE May 2019

Players $A$ and $B$ match pennies $N$ times. They keep a tally of their gains and losses. After the first loss, what is the chance that at no time during the game will they be even?

  1. $\binom{N}{n}/2n$
  2. $\binom{N}{n}/2^{n}$
  3. $\binom{N}{n}/2$
  4. $\binom{N}{n}/2^{N}$
The probability that the running tally is never even (tied) during the game is
$\boxed{\dfrac{\binom{N}{n}}{2^{N}}}$