3.1.1 Calculating Probabilities & Events

Probability is the measure of how likely an event is to occur. Understanding how to calculate probabilities and the relationships between different events is fundamental in statistics.

Basic Probability Concepts

Probability of an Event (P(E))

The probability of an event $E$ is calculated as:

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

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Example: The probability of rolling a $3$ on a fair $6$ -sided die:

P(\text{rolling a 3}) = \frac{1}{6}

Complementary Events

The complement of an event $E$ is the event that $E$ does not occur.

If $P(E)$ is the probability of $E$ , then the probability of the complement $E'$ is:

P(E') = 1 - P(E)

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Example: If $P(E) = 0.4$ ,

then $P(E') = 1 - 0.4 = 0.6$ .

Mutually Exclusive Events

Events are mutually exclusive if they cannot occur at the same time.

If $A$ and $B$ are mutually exclusive, then:

P(A \text{ or } B) = P(A) + P(B)

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Example: The probability of rolling either a $1$ or a $2$ on a $6$ -sided die:

P(\text{rolling a 1 or 2}) = P(1) + P(2)

\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}

Independent Events

Two events are independent if the occurrence of one does not affect the occurrence of the other.

If $A$ and $B$ are independent, then:

P(A \text{ and } B) = P(A) \times P(B)

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Example: The probability of flipping a coin and getting heads, and rolling a die and getting a $4$ :

P(\text{Heads and 4}) = P(\text{Heads}) \times P(4)

\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. The conditional probability of $A$ given $B$ is:

P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}

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Example: In a deck of $52$ $cards$ , what is the probability of drawing an Ace given that a red card has been drawn?

Step 1: Work out the data in the probability question.

There are 26 red cards in the deck ( $13$ $Hearts$ and $13$ $Diamonds$ )

2 of these are Aces.

Step 2: Insert this data into the formula

P(\text{Ace} \mid \text{Red Card}) = \frac{P(\text{Ace and Red Card})}{P(\text{Red Card})}

\frac{\frac{2}{52}}{\frac{26}{52}} = \frac{2}{26} = \frac{1}{13}

Combined Events

Union of Two Events (A or B)

The probability that either event $A$ or event $B$ (or both) occur is given by:

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

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Example: Calculate the probability of drawing a card that is either a heart or an Ace from a deck of cards:

Step 1: Calculate the probability of drawing a heart from a deck of cards.

There are 52 cards in a deck, and 13 of them are hearts.

P(\text{Heart}) = \frac{13}{52} = \frac{1}{4}

Step 2: Calculate the probability of drawing an Ace from a deck of cards.

There are 52 cards in a deck, and 4 of them are Aces.

P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}

Step 3: Calculate the probability of drawing the Ace of Hearts from a deck of cards.

There are 52 cards in a deck, and only one of them is the Ace of Hearts.

$P(\text{Ace of Hearts}) = \frac{1}{52}$

Step 4: Using the formula, calculate the probability of drawing a card that is either a heart or an Ace from a deck of cards:

Use this formula:

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Insert the values from the previous probability calculations.

P(\text{Heart or Ace}) = \frac{1}{4} + \frac{1}{13} - \frac{1}{52}

= \frac{13}{52} + \frac{4}{52} - \frac{1}{52}

= \frac{16}{52} = \frac{4}{13}

Intersection of Two Events (A and B)

The probability that both event $A$ and event $B$ occur is:

P(A \text{ and } B) = P(A) \times P(B \mid A) \quad \text{(if A and B are dependent)}

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Example: Calculate the probability of drawing two Aces in a row without replacement from a deck of cards:

Step 1: Calculate the probability of drawing the first Ace.

There are 52 cards in a deck, and 4 of them are Aces.

Therefore the probability of drawing the first Ace is:

\frac{4}{52} = \frac{1}{13}

Step 2: Calculate the probability of drawing the second Ace.

Since one Ace has already been picked up there are now 51 cards in the deck, with 3 Aces.

Therefore the probability of drawing the second Ace is:

\frac{3}{51} = \frac{1}{17}

Step 3: Using the formula calculate the probability of drawing two Aces in a row without replacement from a deck of cards.

P(\text{Two Aces}) = \frac{1}{13} \times \frac{1}{17} = \frac{1}{221}

Examples of Probability Calculations

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Example 1: Probability in a Dice Game Question: You roll two six-sided dice. What is the probability of getting a sum of $7$ ?

Step 1: Work out the Possible Outcomes that have a sum of 7

Possible outcomes, where the total on both dice adds up to 7: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$

There are 6 favourable outcomes.

Step 2: Work out the Total Possible Outcomes

Total possible outcomes: $6 \times 6 = 36$ .

Step 3: Calculate the Probability

Use this formula:

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

Insert probability values

P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6}

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Example 2: Conditional Probability with Cards Question: What is the probability of drawing a King given that you have drawn a face card ( $Jack, Queen, King$ )?

Step 1: Work out the data needed for the probability question.

There are 12 face cards in a deck (4 Jacks, 4 Queens, 4 Kings).

Step 2: Insert the data into the conditional probability formula

P(\text{King} \mid \text{Face Card}) = \frac{P(\text{King and Face Card})}{P(\text{Face Card})}

Probability of drawing a King given that a face card is drawn:

\frac{4/52}{12/52} = \frac{4}{12} = \frac{1}{3}

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Example 3: Independent Events with Coin Tossing Question: You flip three coins. What is the probability of getting exactly two heads?

Step 1: Work out the possible outcomes of flipping exactly two heads.

Possible outcomes: $HHT, HTH, THH$

3 favourable outcome.

Step 2: Work out the total possible outcomes.

Total possible outcomes: $2^3 = 8$ .

Step 3: Insert the data into the Independent Events formula

P(A \text{ and } B) = P(A) \times P(B)

Using the data from the question:

P(\text{Exactly 2 Heads}) = \frac{3}{8}

Summary

Calculating probabilities involves understanding the nature of the events (independent, mutually exclusive, etc.) and applying appropriate formulas. Whether dealing with simple events, combined events, or conditional probabilities, the key is to carefully analyze the problem, identify all possible outcomes, and calculate the likelihood of each outcome using the rules of probability. The examples provided illustrate how these concepts are applied in different scenarios.

Calculating Probabilities & Events Simplified Revision Notes for A-Level AQA Maths Statistics

3.1.1 Calculating Probabilities & Events

Basic Probability Concepts

Probability of an Event (P(E))

Complementary Events

Mutually Exclusive Events

Independent Events

Conditional Probability

Example: In a deck of $52$ $cards$ , what is the probability of drawing an Ace given that a red card has been drawn?

Combined Events

Union of Two Events (A or B)

Example: Calculate the probability of drawing a card that is either a heart or an Ace from a deck of cards:

Intersection of Two Events (A and B)

Example: Calculate the probability of drawing two Aces in a row without replacement from a deck of cards:

Examples of Probability Calculations

Summary

500K+ Students Use These Powerful Tools to Master Calculating Probabilities & Events For their A-Level Exams.

Other Revision Notes related to Calculating Probabilities & Events you should explore

Calculating Probabilities & Events Simplified Revision Notes for A-Level AQA Maths Statistics

3.1.1 Calculating Probabilities & Events

Basic Probability Concepts

Probability of an Event (P(E))

Complementary Events

Mutually Exclusive Events

Independent Events

Conditional Probability

Example: In a deck of 525252 cardscardscards, what is the probability of drawing an Ace given that a red card has been drawn?

Combined Events

Union of Two Events (A or B)

Example: Calculate the probability of drawing a card that is either a heart or an Ace from a deck of cards:

Intersection of Two Events (A and B)

Example: Calculate the probability of drawing two Aces in a row without replacement from a deck of cards:

Examples of Probability Calculations

Summary

500K+ Students Use These Powerful Tools to Master Calculating Probabilities & Events For their A-Level Exams.

Other Revision Notes related to Calculating Probabilities & Events you should explore

Example: In a deck of $52$ $cards$ , what is the probability of drawing an Ace given that a red card has been drawn?