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Math  /  Data & Statistics

QuestionPart 1 of 4
Catherine plays a game where she draws one card from a well-shuffled standard deck of 52 cards . She wins the game if the card she draws is a King or a Heart.
For this problem, let: - K=\boldsymbol{K}= Catherine draws a King. - ∇=\boldsymbol{\nabla}= Catherine draws a Heart. P(K AND ∇)=P(K \text { AND } \boldsymbol{\nabla})= □\square Submit Part

Studdy Solution

STEP 1

What is this asking? What's the chance Catherine draws a card that's *both* a King *and* a Heart? Watch out! Don't just multiply the chance of a King and the chance of a Heart!
There's a special card that's *both*, and we need to be careful not to count it twice.

STEP 2

1. Calculate the probability of drawing a King.
2. Calculate the probability of drawing a Heart.
3. Calculate the probability of drawing the King of Hearts.
4. Calculate the probability of drawing a King or a Heart.

STEP 3

There are **4** Kings in a standard deck of $52\$52 cards.
So, the probability of drawing a King is: P(K)=452=113P(K) = \frac{\textbf{4}}{\textbf{52}} = \frac{\textbf{1}}{\textbf{13}}

STEP 4

There are **13** Hearts in a standard deck of $52\$52 cards.
So, the probability of drawing a Heart is: P(♡)=1352=14P(\heartsuit) = \frac{\textbf{13}}{\textbf{52}} = \frac{\textbf{1}}{\textbf{4}}

STEP 5

There's only **one** King of Hearts in the deck.
So, the probability of drawing the King of Hearts is: P(K AND ♡)=152P(K \text{ AND } \heartsuit) = \frac{\textbf{1}}{\textbf{52}} This is what the problem is directly asking for!
We're done with the main calculation, but let's go a little further and calculate the chance of drawing *either* a King *or* a Heart.

STEP 6

We might be tempted to just add the probabilities of drawing a King and drawing a Heart, but that would be wrong!
Why? Because we'd be counting the King of Hearts *twice*.
To avoid this, we use the principle of inclusion-exclusion.

STEP 7

The probability of drawing a King *or* a Heart is the sum of the individual probabilities *minus* the probability of drawing both (the King of Hearts): P(K OR ♡)=P(K)+P(♡)−P(K AND ♡)P(K \text{ OR } \heartsuit) = P(K) + P(\heartsuit) - P(K \text{ AND } \heartsuit)

STEP 8

Plugging in the values we calculated earlier: P(K OR ♡)=113+14−152P(K \text{ OR } \heartsuit) = \frac{1}{13} + \frac{1}{4} - \frac{1}{52}

STEP 9

To add these fractions, we need a common denominator, which is **52**: P(K OR ♡)=452+1352−152P(K \text{ OR } \heartsuit) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52}

STEP 10

Now we can add and subtract the numerators: P(K OR ♡)=4+13−152=1652=413P(K \text{ OR } \heartsuit) = \frac{4 + 13 - 1}{52} = \frac{16}{52} = \frac{4}{13}

STEP 11

The probability of drawing the King of Hearts is 152\frac{\textbf{1}}{\textbf{52}}.

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