Basic Probability
In a standard deck of 52 playing cards, what is the probability of drawing a heart?
There are 13 hearts in a deck of 52 cards.
Probability = 13/52 = 1/4 = 0.25 or 25%
A card is drawn from a standard deck. What is P(drawing a heart OR drawing a king)?
Let's solve this step by step:
- P(heart) = 13/52 = 1/4
- P(king) = 4/52 = 1/13
- P(heart AND king) = 1/52 (the king of hearts)
- P(heart OR king) = P(heart) + P(king) - P(heart AND king)
- P(heart OR king) = 1/4 + 1/13 - 1/52 = 13/52 + 4/52 - 1/52 = 16/52 = 4/13
At a school carnival, you flip a fair coin and roll a fair six-sided die. What is the probability of getting both heads and an even number?
Solution:
* Probability of heads = 1/2
* Probability of an even number (2,4,6) = 3/6 = 1/2
* Since events are independent, multiply: 1/2 × 1/2 = 1/4 or 0.25 or 25%
A student has 3 different t-shirts and 2 different pairs of shorts. How many different outfit combinations can they make using one t-shirt and one pair of shorts?
Using the multiplication rule:
Number of combinations = (number of t-shirts) × (number of shorts)
= 3 × 2 = 6 different outfit combinations
What viral food trend of 2023 involved mixing two common breakfast items in a specific way to create "custard toast"?
The trend involved creating a depression in bread, filling it with a mixture of egg, yogurt, and honey, then baking it.
A bag contains 5 red marbles and 3 blue marbles. If you draw two marbles without replacement, what is the probability of drawing two red marbles?
First red: 5/8
Second red: 4/7
Probability = (5/8)(4/7) = 20/56 ≈ 0.357 or about 36%
In a class of 30 students, 12 play basketball and 15 play soccer. If 7 students play both sports, what is the probability that a randomly selected student plays either basketball OR soccer?
Let's solve:
- P(basketball) = 12/30 = 0.4
- P(soccer) = 15/30 = 0.5
- P(both) = 7/30
- P(basketball OR soccer) = P(basketball) + P(soccer) - P(both)
- P(basketball OR soccer) = 0.4 + 0.5 - 7/30 = 0.9 - 7/30 = 20/30 ≈ 0.67
A bag contains 3 red marbles and 4 blue marbles. If you draw two marbles without replacement, what is the probability of drawing two blue marbles?
Solution:
* First blue marble: 4/7
* Second blue marble (now only 3 blue out of 6 total): 3/6
* Multiply since we want both events: 4/7 × 3/6 = 12/42 = 2/7 ≈ 0.286 or 28.6%
A phone password must be 4 digits long, and each digit must be different. How many possible passwords begin with the digit 7?
Let's solve step by step:
- First digit is 7 (1 choice)
- Second digit can be any digit except 7 (9 choices)
- Third digit can be any digit except 7 and the second digit (8 choices)
- Fourth digit can be any remaining digit (7 choices)
= 1 × 9 × 8 × 7 = 504 possible passwords
Which American fast-food restaurant chain went viral on TikTok in 2023 for its "Girl Dinner" marketing campaign featuring small, snack-like combinations?
Dunkin' Donuts embraced the "Girl Dinner" trend with special menu combinations.
Rolling a standard six-sided die, what is the probability of rolling either an even number or a number greater than 4?
Even numbers: {2,4,6}
Numbers > 4: {5,6}
Numbers meeting either condition: {2,4,5,6}
Probability = 4/6 = 2/3 ≈ 0.667 or about 67%
At a tech company, 35% of employees know Python, 42% know Java, and 28% know both languages. What is the probability that a randomly selected employee knows either Python OR Java?
Using the addition rule:
- P(Python OR Java) = P(Python) + P(Java) - P(both)
- P(Python OR Java) = 0.35 + 0.42 - 0.28 = 0.49 or 49%
A student must answer yes or no to two questions on a test. She knows the answer to the first question but guesses on the second. What is the probability she gets both questions correct?
Solution:
* First question (knows it): 1 (100% chance)
* Second question (guessing): 1/2
* Multiply for both events: 1 × 1/2 = 1/2 or 0.5 or 50%
In a school club, students must select a president, vice president, and treasurer. If there are 12 students in the club and no student can hold more than one position, how many different ways can these positions be filled?
Using the multiplication rule:
- President can be any of 12 students
- Vice president can be any remaining 11 students
- Treasurer can be any remaining 10 students
= 12 × 11 × 10 = 1,320 different possible arrangements
The "Tube Girl" trend in 2023 featured people doing what specific activity while riding public transportation?
Recording themselves dancing, lip-syncing, or performing while riding the London Underground (or other subway systems), often to upbeat music.
You have a bag with 3 green balls and 2 yellow balls. What is the probability of drawing a green ball, putting it back, and then drawing a yellow ball?
First draw (green): 3/5
Second draw (yellow): 2/5
Probability = (3/5)(2/5) = 6/25 = 0.24 or 24%
In a survey of 200 students, 120 use Instagram, 85 use Twitter, and 95 use TikTok. If 45 students use Instagram and Twitter, 50 use Instagram and TikTok, 35 use Twitter and TikTok, and 20 students use all three platforms, what is the probability that a randomly selected student uses at least one of these platforms?
Let's solve using the inclusion-exclusion principle:
P(at least one) = P(I) + P(T) + P(Tk) - P(I∩T) - P(I∩Tk) - P(T∩Tk) + P(I∩T∩Tk)
= 120/200 + 85/200 + 95/200 - 45/200 - 50/200 - 35/200 + 20/200
= 0.6 + 0.425 + 0.475 - 0.225 - 0.25 - 0.175 + 0.1
= 0.95 or 95%
Moderate: In a deck of 52 cards, what is the probability of drawing either a king or a heart on a single draw?
Solution:
* Probability of king = 4/52 = 1/13
* Probability of heart = 13/52 = 1/4
* Need to subtract intersection (king of hearts) to avoid double counting
* P(king or heart) = P(king) + P(heart) - P(king of heart)
* = 1/13 + 1/4 - 1/52 = 4/52 + 13/52 - 1/52 = 16/52 = 4/13 ≈ 0.308 or 30.8%
A license plate consists of 2 letters followed by 3 digits. If letters and digits can be repeated, but the first letter cannot be X or Z, how many different license plates are possible?
Let's break it down:
- First letter: 24 choices (26 letters - 2 restricted letters)
- Second letter: 26 choices (any letter)
- Each digit: 10 choices (0-9)
= 24 × 26 × 10 × 10 × 10
= 24 × 26 × 1,000 = 624,000 possible plates
What unique three-word phrase, popularized by creator Alix Earle, became a massive TikTok trend in 2023 for describing getting ready to go out?
"Get Ready With Me" (GRWM) specifically when said as "Ready With Me" in Alix's distinctive style.
A student needs to answer 3 multiple choice questions. Each question has 4 options, and the student guesses randomly. What is the probability of getting at least 2 questions correct?
Let's solve step by step:
- Probability of getting one right = 1/4
- Find probability of getting at least 2 right by:
P(≥2 right) = 1 - [P(0 right) + P(1 right)]
- P(0 right) = (3/4)³ = 27/64
- P(1 right) = C(3,1)(1/4)(3/4)² = 3(1/4)(9/16) = 27/64
- P(≥2 right) = 1 - (27/64 + 27/64) = 1 - 54/64 = 10/64 ≈ 0.156 or about 16%
A manufacturing process has two quality checks. The probability of passing the first check is 0.85, and the probability of passing the second check is 0.78. If the probability of passing both checks is 0.70, what is the probability that a randomly selected product passes at least one of the checks?
Using the addition rule:
- Let A = passing first check, B = passing second check
- P(A OR B) = P(A) + P(B) - P(A AND B)
- P(A OR B) = 0.85 + 0.78 - 0.70 = 0.93 or 93%
A jar contains 5 white, 3 black, and 2 red marbles. Two marbles are drawn without replacement. What is the probability of drawing either two white marbles or two marbles of different colors?
Solution:
* P(two white) = 5/10 × 4/9 = 20/90
* P(different colors) =
* P(white then black) = 5/10 × 3/9 +
* P(black then white) = 3/10 × 5/9 +
* P(white then red) = 5/10 × 2/9 +
* P(red then white) = 2/10 × 5/9 +
* P(black then red) = 3/10 × 2/9 +
* P(red then black) = 2/10 × 3/9
* = 20/90 + (15/90 + 15/90 + 10/90 + 10/90 + 6/90 + 6/90)
* = 20/90 + 62/90 = 82/90 ≈ 0.911 or 91.1%
At a restaurant, a meal consists of:
- One appetizer (6 choices)
- One main course (8 choices)
- One dessert (5 choices)
- One beverage (4 choices)
If a customer wants to order exactly three items from these categories (but it doesn't matter which three categories they choose), how many different possible combinations are there?
Let's solve step by step:
1) First, identify all possible ways to choose 3 categories:
- Appetizer, Main, Dessert
- Appetizer, Main, Beverage
- Appetizer, Dessert, Beverage
- Main, Dessert, Beverage
2) Calculate each possibility:
- Appetizer, Main, Dessert: 6 × 8 × 5 = 240
- Appetizer, Main, Beverage: 6 × 8 × 4 = 192
- Appetizer, Dessert, Beverage: 6 × 5 × 4 = 120
- Main, Dessert, Beverage: 8 × 5 × 4 = 160
3) Total combinations = 240 + 192 + 120 + 160 = 712 possible combinations
The "Roman Empire" trend of 2023 revealed what surprising fact about many men's thought patterns, leading to viral videos asking partners how often they think about what historical topic?
The trend revealed that many men think about the Roman Empire surprisingly frequently (daily or weekly), leading to viral videos of women asking their male partners about this phenomenon.