What is the probability of getting a spade on the first draw and a heart on the second?

 

To calculate the probability of drawing a spade on the first draw and a heart on the second draw from a standard deck of 52 cards, we can follow these steps:

1. Calculate the probability of drawing a spade on the first draw:

   • There are 13 spades in a standard deck of 52 cards.
   • Therefore, the probability of drawing a spade on the first draw is $\frac{13}{52}$.

2. Calculate the probability of drawing a heart on the second draw, given that the first card drawn was a spade:

   • After drawing a spade, there are now 51 cards left in the deck.
   • There are still 13 hearts in the deck.
   • Therefore, the probability of drawing a heart on the second draw is $\frac{13}{51}$.

3. Calculate the combined probability of both events happening:

   • The probability of both independent events happening is the product of their individual probabilities.

Thus, the probability of drawing a spade on the first draw and a heart on the second draw is:

$$\frac{13}{52} \times \frac{13}{51} = \frac{1}{4} \times \frac{13}{51} = \frac{13}{204}$$

Therefore, the probability is $\frac{13}{204}$.

Let's continue by simplifying the fraction to get the final probability.

We have:

$$\frac{13}{204}$$

To simplify this fraction, we check if the numerator and the denominator have any common factors. The numerator is 13, which is a prime number, meaning it only has 1 and 13 as factors.

The denominator is 204. Let's check if 13 is a factor of 204:

$$204 \div 13 \approx 15.69$$

Since 15.69 is not an integer, 13 is not a factor of 204. Therefore, $\frac{13}{204}$ is already in its simplest form.

So, the final simplified probability of drawing a spade on the first draw and a heart on the second draw is:

$$\frac{13}{204}$$

This fraction can also be expressed as a decimal:

$$\frac{13}{204} \approx 0.0637$$

Therefore, the probability is approximately 0.0637, or 6.37%.