Equally Likely Outcomes and Counting Probability

From Equally Likely Outcomes to Probability

In Sample Spaces and Outcomes, we learned to describe a chance process by listing its possible outcomes. In Events as Subsets of the Sample Space, we learned that an event consists of outcomes that meet a stated condition. Now we can calculate probabilities for a particularly useful kind of chance model: one in which every individual outcome is equally likely.

Consider drawing one card from a well-shuffled standard deck. The deck has 52 cards: 13 ranks in each of four suits. Unless stated otherwise, it has no jokers. If the deck is well shuffled and one card is drawn, each of those 52 individual cards has the same probability of being selected. A probability can therefore be found by counting how many individual cards satisfy the event and comparing that count with 52.

A favorable outcome is an outcome in the event being considered. “Favorable” does not mean that the outcome is good or desirable; it means only that it meets the event’s definition. For a one-card draw, the individual cards are the outcomes. If the event is “draw a heart,” each heart is one favorable outcome.

The denominator comes from the entire sample space for the chance process—not just the cards that might seem relevant to the event. For one draw from a standard deck, there are 52 possible individual-card outcomes, so the denominator is 52. The numerator counts how many of those 52 cards belong to the event.

Count the Individual Cards

It helps to know how the 52 cards are organized. There are four suits—hearts, diamonds, clubs, and spades—with 13 cards in each suit. Hearts and diamonds are red; clubs and spades are black. The ranks are ace, 2 through 10, jack, queen, and king. In this tutorial, a face card means a jack, queen, or king. An ace is not a face card under this definition.

These counts are shortcuts for organizing the outcomes, but they do not change what is being counted: individual cards. For example, “a queen” is an event containing four cards, one queen in each suit. The event’s probability is \(4/52\), not \(1/13\) because “queen” is one rank label among 13. Both calculations happen to agree after simplifying: \(4/52=1/13\). The favorable-over-total count makes clear why the probability is correct.

A probability may be left as a fraction or written as a decimal or percentage, as appropriate. Simplifying a fraction does not change its value. A decimal or percentage is often rounded, so indicate that rounding when the number is not exact.

Use the Event to Decide What Counts

The wording of an event determines which cards count as favorable. “A king” means any of the four kings. “A black king” means only the king of clubs and the king of spades. A card that does not meet the event’s condition is not favorable, even if it shares one feature with cards that do.

It can be useful to write down the event in ordinary language before counting. Then identify the relevant cards by suit, rank, color, or a combination of those characteristics. This prevents a common counting error: using a correct-looking count for the wrong event.

When Event Categories Overlap

Some events can be described using the word “or,” as in “a red card or a queen.” A card can meet both parts of that description. The queen of hearts and queen of diamonds are both red cards and queens. If we count all 26 red cards and then add all four queens, those two cards have been counted twice.

When two event categories overlap, count each individual outcome only once in the event. One practical method is to count the outcomes that meet the first condition, then add only the outcomes that meet the second condition but were not already included. Another is to count the individual cards in the combined event directly. Both methods avoid double-counting.

A Reliable Counting Routine

For a card-drawing question with equally likely outcomes, use a short routine to keep the event, sample space, and calculation connected. This is a counting method, not a new chance process: the sample space is still the set of individual cards that could be drawn.

This routine also offers a quick check: the favorable count cannot be greater than the total count. The resulting probability should be between 0 and 1, as described in Probability Is a Number Between 0 and 1. If a count or probability falls outside those limits, revisit the event definition and the counting.

Common Mistakes and AP Exam Tips

• Using the number of categories as the denominator. There are four suits, but a one-card draw has 52 individual outcomes. Use 52 as the denominator for a single draw from a standard deck.
• Counting the wrong event. There are 12 face cards in all, but only six black face cards. Match the favorable count to every condition in the event.
• Double-counting overlap. In “red or queen,” the two red queens meet both conditions. Count each card once; do not treat overlapping descriptions as separate card outcomes.
• Confusing an exact fraction with a rounded decimal. The probability \(7/13\) is exact, while 0.5385 is a rounded decimal approximation. Keep the exact fraction when it communicates the answer clearly.
• Assuming the counting formula always applies. Favorable outcomes divided by total outcomes works in this form because the 52 individual cards are equally likely under the stated well-shuffled-deck model. If outcomes are not equally likely, a simple count ratio may not give the probability.
• Giving a number without naming the event. A complete answer identifies what the probability is about—for example, “The probability of drawing a black face card is \(3/26\).”

On an AP response, show the count as well as the ratio. Writing “six black face cards out of 52 cards” makes the sample space and the favorable outcomes visible, so a reader can check how the probability was obtained. Then give the probability in context and label any rounded decimal as approximate.