Questions: When choosing one card from a standard 52 -card deck, find the probability of drawing a 2 or a red card? The answer must be represented as a reduced fraction.

Solution Steps

To find the probability of drawing a 2 or a red card from a standard 52-card deck, we need to consider the following:

1. Calculate the probability of drawing a 2. There are 4 twos in a deck (one for each suit).
2. Calculate the probability of drawing a red card. There are 26 red cards in a deck (13 hearts and 13 diamonds).
3. Use the principle of inclusion-exclusion to avoid double-counting the red twos. There are 2 red twos (2 of hearts and 2 of diamonds).
4. Combine these probabilities to find the total probability of drawing a 2 or a red card.

In a standard 52-card deck, there are 4 twos (one for each suit). Therefore, the probability of drawing a 2 is given by:

\[ P(2) = \frac{4}{52} = \frac{1}{13} \]

There are 26 red cards in a standard deck (13 hearts and 13 diamonds). Thus, the probability of drawing a red card is:

\[ P(\text{Red}) = \frac{26}{52} = \frac{1}{2} \]

Among the red cards, there are 2 red twos (2 of hearts and 2 of diamonds). Therefore, the probability of drawing a red 2 is:

\[ P(\text{Red 2}) = \frac{2}{52} = \frac{1}{26} \]

To find the probability of drawing a 2 or a red card, we use the inclusion-exclusion principle:

\[ P(2 \cup \text{Red}) = P(2) + P(\text{Red}) - P(\text{Red 2} \]

Substituting the values we calculated:

\[ P(2 \cup \text{Red}) = \frac{1}{13} + \frac{1}{2} - \frac{1}{26} \]

To combine these fractions, we find a common denominator, which is 26:

\[ P(2 \cup \text{Red}) = \frac{2}{26} + \frac{13}{26} - \frac{1}{26} = \frac{2 + 13 - 1}{26} = \frac{14}{26} = \frac{7}{13} \]

Final Answer