Find the Row for One Exact Value
In Probability Distribution of a Discrete Random Variable, you learned that a distribution table pairs each possible value of a random variable with its probability. In Drawing a Probability Histogram and Describing Shape Center and Spread of a Distribution, you used those probabilities to understand the distribution as a whole. This tutorial focuses on one direct question: what is the probability that the random variable takes one particular value?
The notation \(P(X=x)\) means “the probability that the random variable \(X\) equals the value \(x\).” The capital letter \(X\) names the random variable; the lowercase \(x\) stands for one specific numerical value it could take. To find \(P(X=x)\) from a distribution table, locate the row where the value is \(x\), then read the probability in that row.
“Exact value” matters. If \(X\) counts how many items are damaged, then \(P(X=2)\) refers to exactly 2 damaged items—not 1 or 3, and not “2 or more.” A table entry is attached to one value of the variable. The wording in your interpretation should keep that same exact-value meaning.
A probability is a number from 0 to 1, while \(x\) is a possible value of the random variable. For example, if \(P(X=2)=0.18\), then 2 is the value and 0.18 is its probability. You can express 0.18 as 18%, but you should not say that the variable equals 0.18. The variable equals 2 with probability 0.18.
Say What the Probability Means
A numerical answer is only part of a good response. Interpret the probability in the setting by naming what \(X\) represents and what the exact value \(x\) means. A clear sentence follows this pattern: “The probability that [the random variable’s meaning] is exactly [value] is [probability].”
For example, if \(X\) is the number of late buses arriving at a stop during a morning, then \(P(X=1)=0.24\) means that the model assigns a probability of 0.24, or 24%, to exactly one late bus arriving during that morning. It does not mean that 24% of buses are late, because the random variable counts late buses per morning rather than describing an individual bus.
A probability describes the chance assigned by the model to an outcome of the chance process. It does not promise what will happen on one particular occasion. If the same well-defined chance process is repeated many times under the model, the proportion of repetitions with \(X=x\) would be expected to be approximately \(P(X=x)\). That long-run idea helps interpret the probability, but it does not change which exact event the table entry describes.
Worked Example: Read a Bike-Share Return Count
Worked Example: Read a Bike-Share Return Count
For an invented model, let \(X\) be the number of bicycles returned to a small station during a specified 15-minute interval. The distribution table is:
| Bicycles returned, \(x\) | \(P(X=x)\) |
|---|---|
| 0 | 0.08 |
| 1 | 0.22 |
| 2 | 0.36 |
| 3 | 0.24 |
| 4 | 0.10 |
State. Find and interpret \(P(X=2)\).
Plan. The question asks for the probability of the exact value 2. Find the row where the number of bicycles returned is 2, then read the probability in that row and describe the event in context.
Do. The row for \(x=2\) lists \(P(X=2)=0.36\). As a percentage, \(0.36 \times 100\%=36\%\).
Conclude. According to this model, the probability that exactly 2 bicycles are returned to the station during the specified 15-minute interval is 0.36, or 36%.
The event is not “at least 2 bicycles” or “no more than 2 bicycles.” The table entry is for exactly 2. Keeping that word in the conclusion makes the interpretation match the notation.
Match the Notation to the Row
When a question gives notation such as \(P(Y=3)\), first identify what \(Y\) counts or measures. Then identify the specific value after the equals sign. The table may label its first column with the random variable’s values and its second column with \(P(Y=y)\), or it may use wording such as “number of repairs” and “probability.” In either case, use the value column to find the correct row.
Be careful not to mistake a table heading or a value for the probability itself. The value \(y=3\) might be paired with probability 0.15. The requested probability is 0.15, not 3. Conversely, if a question asks which value has probability 0.15, look across the table to find the value paired with that probability.
If a table gives every possible value of a discrete random variable, a value outside the listed possibilities cannot occur in that model and has probability 0. But do not assume a value is impossible just because it is missing from a partial display; first determine whether the table is meant to show the entire distribution. In a complete distribution table, every possible value has a corresponding probability.
Worked Example: Interpret a Quality-Check Result
Suppose an invented quality-control model uses \(D\) for the number of small surface marks on a randomly selected ceramic tile. Its probability distribution is:
| Surface marks, \(d\) | \(P(D=d)\) |
|---|---|
| 0 | 0.52 |
| 1 | 0.31 |
| 2 | 0.13 |
| 3 | 0.04 |
State. Find and interpret the probability that the selected tile has exactly 3 surface marks.
Plan. Translate “exactly 3 surface marks” into \(D=3\). Read the probability from the row with \(d=3\), then state what that probability means for one randomly selected tile under this model.
Do. The row for \(d=3\) gives \(P(D=3)=0.04\). Converting to a percentage gives \(0.04 \times 100\%=4\%\).
Conclude. Under the model, the probability that a randomly selected tile has exactly 3 small surface marks is 0.04, or 4%.
The result is about a tile’s count of marks. It is not the probability that a tile has “at least 3 marks,” and it is not a claim that every group of 100 tiles will contain exactly 4 tiles with 3 marks.
Compare Single-Value Probabilities Carefully
A table can answer several exact-value questions. To compare two values, read each of their probabilities and describe which exact event is more likely according to the model. A larger probability means that one value is more likely than the other, but it does not mean that value is guaranteed to occur.
If two values have equal probabilities, neither exact value is more likely under the model. And if a value has the largest probability in the table, it is the most likely single value, or mode, as discussed in Describing Shape Center and Spread of a Distribution. That still does not tell you the probability of a broader event involving multiple values. Keep the question focused on the requested single value.
Worked Example: Compare Package-Delay Counts
In another invented model, let \(K\) be the number of packages delayed during a small delivery route on a given day. The model gives this distribution:
| Packages delayed, \(k\) | \(P(K=k)\) |
|---|---|
| 0 | 0.16 |
| 1 | 0.34 |
| 2 | 0.30 |
| 3 | 0.14 |
| 4 | 0.06 |
State. Which is more likely according to the model: exactly 1 delayed package or exactly 3 delayed packages? Give both probabilities and interpret the comparison.
Plan. Read the entries for \(K=1\) and \(K=3\), compare them, and describe the two exact events in the context of one delivery route on one day.
Do. The table gives \(P(K=1)=0.34\) and \(P(K=3)=0.14\). Since \(0.34>0.14\), the exact outcome of 1 delayed package has the larger probability. In percentage form, the probabilities are 34% and 14%.
Conclude. According to the model, exactly 1 package being delayed on a given route-day is more likely than exactly 3 packages being delayed: their probabilities are 0.34 and 0.14, respectively. This comparison concerns those two exact counts only.
Common Mistakes and AP Exam Tips
- Giving the value instead of the probability. If asked for \(P(X=2)\), “2” is not the answer. Find the probability paired with 2 and report that number.
- Dropping the word “exactly.” \(P(X=2)\) means that \(X\) equals 2. A full-credit interpretation says “exactly 2” and names what \(X\) counts or measures.
- Mixing up the variable and its value. \(X\) names the random variable; \(x\) is one possible value. Explain what the variable represents before interpreting a particular entry.
- Changing decimals into percentages incorrectly. Multiply a probability by 100 to express it as a percentage: 0.07 is 7%, not 0.7% or 70%.
- Claiming the outcome is certain from its probability. A probability describes the model’s chance for the event; it does not guarantee what will happen in one repetition. Avoid saying an event “will happen” just because its probability is positive.
- Treating a probability table as a count table. A probability such as 0.20 is a model probability, not automatically 20 observed cases. Read the column heading before explaining an entry.
For an AP response, show the lookup when useful, write the probability with the correct notation, and finish with a complete sentence in context. A strong interpretation identifies the random variable, states the exact value, and attaches the table probability to that event.
Key Takeaway
A discrete distribution table gives the probability for every possible value of its random variable. Answer a single-value question by matching the requested value to its row, then explain what that exact outcome means in the situation.
Check Your Understanding
Use the distribution below for \(R\), the number of reusable water bottles sold at a pop-up stand during a specified hour.
| Bottles sold, \(r\) | \(P(R=r)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.28 |
| 2 | 0.35 |
| 3 | 0.18 |
| 4 | 0.07 |
- Find \(P(R=1)\) and interpret it in context.
- What exact event is represented by \(P(R=3)\)?
- Express \(P(R=4)\) as a percentage.
- According to the model, is exactly 2 bottles sold more likely than exactly 0 bottles sold? Support your answer with the table entries.
- Explain why \(P(R=2)\) does not mean that the number of bottles sold is 0.35.