Probability of Selecting a Number Less Than 9 from a Deck of Ten Cards

Understanding the Given Probability Problem

This question is based on the fundamental concept of probability. A deck contains ten cards, and each card has a different number from 1 to 10. One card is selected randomly, which means that every card has an equal chance of being drawn.

The numbers written on the ten cards are:

1, 2, 3, 4, 5, 6, 7, 8, 9, and 10

We need to calculate the probability that the number written on the selected card is less than 9. To solve this problem, we first identify the total number of possible outcomes and then count the number of outcomes that satisfy the required condition.

Basic Formula of Probability

When all outcomes of an experiment are equally likely, the probability of an event is calculated using the following formula:

Probability of an event = Number of favorable outcomes / Total number of possible outcomes

In this question, a favorable outcome means selecting a card carrying a number less than 9. The total number of possible outcomes is equal to the total number of cards in the deck.

Step-by-Step Solution

Step 1: Determine the Total Number of Possible Outcomes

The deck contains ten cards numbered from 1 to 10. Since one card is drawn at random, any one of these ten cards can be selected.

Therefore:

Total number of possible outcomes = 10

The complete sample space can be written as:

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Thus, the number of elements in the sample space is 10.

Step 2: Identify the Numbers Less Than 9

The question asks for the probability of selecting a number less than 9. The phrase “less than 9” means that the selected number must be strictly smaller than 9. Therefore, the number 9 itself is not included.

The numbers less than 9 in the given deck are:

1, 2, 3, 4, 5, 6, 7, and 8

There are eight such cards. Therefore:

Number of favorable outcomes = 8

Step 3: Apply the Probability Formula

Using the basic probability formula:

Probability = Number of favorable outcomes / Total number of possible outcomes

Substituting the values obtained above:

P(number less than 9) = 8 / 10

Simplifying the fraction:

P(number less than 9) = 4 / 5

Converting the fraction into decimal form:

P(number less than 9) = 0.8

Step 4: Express the Answer to One Decimal Place

The question specifically asks for the answer to one decimal place. The calculated probability is already 0.8, which contains exactly one digit after the decimal point.

Therefore:

Probability = 0.8

Why 9 Is Not Included Among the Favorable Outcomes

The wording of the question is important. It asks for a number less than 9, not a number less than or equal to 9. Therefore, only the numbers from 1 to 8 satisfy the given condition.

If the question had asked for a number less than or equal to 9, then the card numbered 9 would also have been included, giving nine favorable outcomes. However, in the present question, the number 9 must be excluded.

Understanding the Answer Through Probability Range

The probability of any event always lies between 0 and 1. A probability of 0 represents an impossible event, while a probability of 1 represents a certain event.

Here, eight out of the ten cards contain numbers less than 9. Therefore, the probability is 0.8, which means there is an 80% chance of selecting a number less than 9.

This result is reasonable because most of the cards in the deck satisfy the required condition. Only two cards, numbered 9 and 10, do not contain numbers less than 9.

Alternative Method Using the Complement Rule

The same answer can also be obtained using the complement rule of probability. Instead of directly counting the cards numbered less than 9, we can count the cards that do not satisfy the condition.

The numbers that are not less than 9 are:

9 and 10

Therefore, the probability of selecting a number that is not less than 9 is:

2 / 10 = 0.2

Using the complement rule:

P(number less than 9) = 1 − P(number not less than 9)

Therefore:

P(number less than 9) = 1 − 0.2

P(number less than 9) = 0.8

This alternative calculation confirms the result obtained using the direct probability method.

Final Answer

The probability of selecting a number less than 9 from the deck of ten cards is 0.8.