Probability Problems: Single Events

This section focuses on understanding and calculating probabilities related to single events. These events involve situations with a defined set of possible outcomes, such as flipping a coin, rolling a die, or drawing a card from a deck. The core concept revolves around the likelihood of a specific outcome occurring.

Probability is quantified as a value between 0 and 1, inclusive. A probability of 0 indicates an impossible event, while a probability of 1 signifies a certain event. The closer the probability is to 1, the more likely the event is to occur.

We calculate probability by dividing the number of favorable outcomes (the outcomes we’re interested in) by the total number of possible outcomes.

Formulae

The fundamental formula for calculating the probability of a single event is:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Where:

• $P(E)$ represents the probability of event E.
• ‘Favorable outcomes’ are the specific outcomes we are interested in.
• ‘Total number of possible outcomes’ is the size of the sample space.

Examples

Example-1: Coin Toss

What is the probability of getting heads when flipping a fair coin?

Solution:

There is one favorable outcome (heads) and two possible outcomes (heads or tails).

So, $P(\text{Heads}) = \frac{1}{2} = 0.5$ or 50%.

Example-2: Rolling a Die

What is the probability of rolling an even number on a standard six-sided die?

Solution:

There are three favorable outcomes (2, 4, and 6) and six possible outcomes (1, 2, 3, 4, 5, and 6).

So, $P(\text{Even number}) = \frac{3}{6} = \frac{1}{2} = 0.5$ or 50%.

Common mistakes by students

• Incorrectly identifying favorable outcomes: Failing to accurately determine the outcomes that satisfy the event’s criteria.
• Miscounting total outcomes: Making errors in counting all possible outcomes, often due to overlooking a possibility or double-counting.
• Confusing probability with odds: Probability is the ratio of favorable outcomes to all possible outcomes, whereas odds are the ratio of favorable outcomes to unfavorable outcomes. Students might confuse these two related but distinct concepts.
• Using the wrong formula: Applying a more complex formula for independent events when dealing with a single event, unnecessarily complicating the calculation.

Real Life Application

Understanding probabilities of single events is useful in many aspects of real life.

• Gambling: Calculating the probability of winning in games like poker, or roulette.
• Insurance: Insurance companies use probability to assess the risk of events like car accidents or house fires, allowing them to set premiums.
• Weather forecasting: Predicting the likelihood of rain or other weather events.
• Medical diagnostics: Doctors can use probabilities to interpret test results and estimate the chance of a particular disease.

Fun Fact

The study of probability theory has deep historical roots. Mathematicians like Blaise Pascal and Pierre de Fermat were pioneers in the field, developing key concepts while trying to solve gambling problems in the 17th century. Their work laid the foundation for modern probability and statistics!

Recommended YouTube Videos for Deeper Understanding

Q.1 A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

Check Solution

Ans: B

The numbers greater than 4 on a die are 5 and 6. There are 2 favorable outcomes and 6 total possible outcomes. The probability is $\frac{2}{6} = \frac{1}{3}$.

Q.2 A standard deck of 52 playing cards is shuffled. What is the probability of drawing a King?

Check Solution

Ans: B

There are 4 Kings in a standard deck of 52 cards. The probability is $\frac{4}{52} = \frac{1}{13}$.

Q.3 A coin is flipped twice. What is the probability of getting heads on both flips?

Check Solution

Ans: A

The possible outcomes are HH, HT, TH, TT. Only one outcome (HH) results in heads on both flips. The probability is $\frac{1}{4}$.

Q.4 A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of picking a blue marble?

Check Solution

Ans: B

There are 3 blue marbles and a total of 10 marbles (5+3+2). The probability is $\frac{3}{10}$.

Q.5 A single card is drawn from a standard deck of 52 cards. What is the probability of drawing a card that is either a Queen or a club?

Check Solution

Ans: D

There are 4 Queens and 13 clubs. However, the Queen of clubs is counted in both. There are 4 Queens + 13 Clubs – 1 Queen of clubs = 16 favorable outcomes. The probability is $\frac{16}{52}=\frac{4}{13}$.

Next Topic: Rational Numbers: Representation & Decimal Expansions