Question

If the first four cards drawn from the deck are a king, a queen, a jack, and a 10 , what is the probability that the fifth card drawn will be an ace? A. \(\frac{1}{12}\) B. \(\frac{4}{13}\) C. \(\frac{4}{51}\) D. \(\frac{12}{13}\)

Short answer

A. \(\frac{1}{12}\)

Step-by-step solution

Understand the problem

The problem is to find the probability that the fifth card drawn from a deck of 52 cards will be an ace, given that the first four cards drawn were a king, a queen, a jack, and a 10.

Determine the remaining cards

After drawing a king, queen, jack, and 10, there are 48 cards left in the deck (52 - 4 = 48 cards).

Identify the number of favorable outcomes

There are 4 aces in a standard deck of 52 cards. None of the aces have been drawn yet, so all 4 aces remain in the deck.

Calculate the probability

The probability that the fifth card drawn is an ace is given by the ratio of the number of favorable outcomes to the number of remaining cards. Thus, the probability is \(\frac{4}{48} = \frac{1}{12}\).

Key concepts

probability calculation

Probability is a way to measure the likelihood of an event occurring. To calculate probability, we use the ratio of the number of favorable outcomes to the total number of possible outcomes.
For example, in this exercise, we are asked to find the probability that the fifth card drawn from a deck of cards is an ace.
We start by noting that the favorable outcomes (drawing an ace) need to be counted against the total outcomes (remaining cards in the deck).
The formula used here would be \( P(E) = \frac{number\ of\ favorable\ outcomes}{total\ number\ of\ possible\ outcomes} \). So, if we have all the favorable outcomes and the total outcomes correctly identified, calculating the probability becomes straightforward.

standard deck of cards

A standard deck of cards is a set of 52 cards divided into 4 suits: hearts, diamonds, clubs, and spades.
Each suit contains 13 cards: ace, 2 through 10, jack, queen, and king. Knowing how a deck is structured helps in determining the number of possible and favorable outcomes.
In this exercise, the first step we took was noting that there are 48 remaining cards after the first four were drawn.
This is because we know a standard deck always starts with 52 cards. As we draw more cards, the total number of cards decreases, giving us our 'total number of possible outcomes'.

favorable outcomes

Favorable outcomes are specific outcomes we are interested in for our probability calculation.
In this case, we are interested in drawing an ace. We start by identifying all aces in the deck.
Since none of the aces were drawn in the first four cards, all 4 aces remain.
Therefore, the number of favorable outcomes is 4. This directly impacts our probability calculation.
It is always important to correctly count the favorable outcomes to ensure the accuracy of your probability result.

ratio in probabilities

The ratio in probability tells us how likely it is for an event to happen compared to all possible outcomes.
To express this, we use ratios or fractions. For example, the probability formula provided earlier tells us the likelihood of drawing an ace as the 5th card using a ratio of favorable to possible outcomes.
In this problem, after counting the favorable outcomes (4 aces remaining) and total possible outcomes (48 remaining cards), we apply the ratio \( \frac{4}{48} \).
Simplifying this ratio, we get \( \frac{1}{12} \), which means there is a 1 in 12 chance of drawing an ace as the fifth card.