Finding card draw probabilities from a mixed deck table

Key concept: This is a counting-and-probability question. The solution depends on identifying the total number of cards, counting each event correctly, and expressing the probability as a fraction, decimal, and percent.

Step by step

Read the sample space first

The list of cards shown in the problem is the complete sample space. Before answering any probability question, count the total number of outcomes. In this case, there are 12 cards, so every probability is based on a denominator of 12.

That is the first skill this problem tests: probability is always

$$P(\text{event})=\frac{\text{number of favorable outcomes}}{\text{total outcomes}}.$$

Count each event carefully

For the card labeled $5$, there is exactly one $5$ in the list, so

$$P(5)=\frac{1}{12}.$$

For the letter $J$, there are two J cards, so

$$P(J)=\frac{2}{12}=\frac{1}{6}.$$

For a number, count every numeric card in the deck. The list contains eight numbers, so

$$P(\text{a number})=\frac{8}{12}=\frac{2}{3}.$$

For the card $4$, there are two cards labeled $4$, so

$$P(4)=\frac{2}{12}=\frac{1}{6}.$$

Check the category wording

The event $P(T)$ is zero because there is no card labeled $T$ in the list. A probability of zero means the event is impossible, not merely unlikely.

For a letter, count all letter cards. The list contains four letters, so

$$P(\text{a letter})=\frac{4}{12}=\frac{1}{3}.$$

That makes the probability table consistent: numbers and letters partition the deck, and each count must match the actual visible cards.

Final answers in the same format

• $P(5)=\frac{1}{12}=0.083=8.3\%$
• $P(J)=\frac{2}{12}=0.167=16.7\%$
• $P(\text{a number})=\frac{8}{12}=0.333=33.3\%$
• $P(4)=\frac{2}{12}=0.167=16.7\%$
• $P(T)=0$
• $P(\text{a letter})=\frac{4}{12}=0.333=33.3\%$

Pitfall alert

The most common mistake is counting repeated values separately only when convenient but ignoring them in other events. Every occurrence matters. If there are two J cards, both belong in the count for $P(J)$. Another mistake is treating "a number" as if it excludes repeated numeric symbols or includes letters by accident. Always define the event first, then count only the cards that satisfy it. Also, if a symbol does not appear at all, its probability is exactly zero, not a small decimal approximation.

Try different conditions

If the deck changed to include an extra card, the probabilities would all need to be recomputed from the new total. For example, if the question became: "One more card labeled 7 is added. What is $P(7)$ and $P(a number)$?" then the denominator would increase by 1, the count for 7 would rise by 1, and every percent would change. This is why probability questions must always start with the updated sample space, not with a memorized fraction.

Further reading

sample space, theoretical probability, favorable outcomes