Question

A standard deck of 52 cards contains four suits: clubs, spades, hearts, and diamonds. Each deck contains an equal number of cards in each suit. Rochelle chooses a card from the deck, records the suit, and replaces the card. Her results are shown in the table.

Cards
Suit
Observed Frequency
Clubs
29
Spades
13
Hearts
15
Diamonds
23

How does the experimental probability of choosing a heart compare with the theoretical probability of choosing a heart?
The theoretical probability of choosing a heart is StartFraction 1 over 16 EndFraction greater than the experimental probability of choosing a heart.
The experimental probability of choosing a heart is StartFraction 1 over 16 EndFraction greater than the theoretical probability of choosing a heart.
The theoretical probability of choosing a heart is StartFraction 1 over 26 EndFraction greater than the experimental probability of choosing a heart.
The experimental probability of choosing a heart is StartFraction 1 over 26 EndFraction greater than the theoretical probability of choosing a heart.

Answer 1

To determine how the experimental probability of choosing a heart compares with the theoretical probability, we need to calculate both probabilities.

**Theoretical Probability:**
In a standard deck of 52 cards, there are 4 suits, each containing 13 cards. Therefore, the theoretical probability of choosing a heart (P(Heart)) is:

\[
P(\text{Heart}) = \frac{\text{Number of Hearts}}{\text{Total Number of Cards}} = \frac{13}{52} = \frac{1}{4}
\]

**Experimental Probability:**
Now, let's compute the experimental probability based on Rochelle's observed frequencies. The total number of cards she observed is the sum of all the observed frequencies:

\[
\text{Total Observations} = 29 + 13 + 15 + 23 = 80
\]

The observed frequency of hearts is 15. Therefore, the experimental probability of choosing a heart (P(Heart)) is:

\[
P(\text{Heart}) = \frac{\text{Observed Frequency of Hearts}}{\text{Total Observations}} = \frac{15}{80}
\]

To simplify \( \frac{15}{80} \):

\[
\frac{15}{80} = \frac{3}{16}
\]

**Comparison:**
Now we need to compare \( \frac{3}{16} \) (experimental probability) with \( \frac{1}{4} \) (theoretical probability). To make this easier, we'll convert \( \frac{1}{4} \) to a fraction with a denominator of 16:

\[
\frac{1}{4} = \frac{4}{16}
\]

Now we can compare the two probabilities:

- Experimental Probability (E.P) = \( \frac{3}{16} \)
- Theoretical Probability (T.P) = \( \frac{4}{16} \)

Now, we can see that:

\[
\frac{4}{16} - \frac{3}{16} = \frac{1}{16}
\]

This shows that the theoretical probability is \( \frac{1}{16} \) greater than the experimental probability.

So, the correct answer is:
**The theoretical probability of choosing a heart is \( \frac{1}{16} \) greater than the experimental probability of choosing a heart.**