RBSE Solutions Class 11 Maths Chapter 14 Probability Exercise 14.1

Question 1. From a carton of bulbs, 3 bulbs are drawn randomly. Test each bulb and classified into damage (D) and not damage (N). Find the sample space of this test.
Answer: When drawing the first bulb, there are two possible results: Damaged (D) or Not Damaged (N).
When drawing the second bulb, each result from the first bulb can combine with D or N. This is represented as \( (D,N) \times (D,N) \). This forms the sample space for two bulbs, which is \( \{DD, DN, ND, NN\} \).
Next, for the third bulb, each outcome from the two bulbs can again combine with either D or N. This means we are effectively combining \( (D,N) \) with each element of \( \{DD, DN, ND, NN\} \). The total number of possible outcomes will be \( 2 \times 2 \times 2 = 8 \) because each of the three bulbs has two distinct possibilities.
Thus, the full sample space \( S \) for this test, listing all combinations of 3 bulbs, is:
\( S = \{DDD, DDN, DND, DNN, NDD, NDN, NND, NNN\} \)
In simple words: When you pick three bulbs, each one can either be damaged (D) or not damaged (N). The sample space is a list of all the different ways these three bulbs can turn out, like all damaged (DDD) or all not damaged (NNN).

🎯 Exam Tip: When listing outcomes for multiple events, use a tree diagram or systematic multiplication of possibilities to ensure all combinations are included and none are repeated.

Question 3. A dice is thrown E is an event to getting 4 on dice and getting even number is event F. Are the events E and F mutually exclusive events?
Answer: When a dice is thrown, the possible results, called the sample space \( S \), are \( \{1, 2, 3, 4, 5, 6\} \).
According to the question, event \( E \) is getting a 4, so \( E = \{4\} \).
Event \( F \) is getting an even number, so \( F = \{2, 4, 6\} \).
For events to be mutually exclusive, they must not have any outcomes in common; their intersection must be empty. Mutually exclusive events cannot occur simultaneously. Here, the intersection of \( E \) and \( F \) is \( E \cap F = \{4\} \).
Since \( \{4\} \) is not an empty set (it contains the number 4), events \( E \) and \( F \) are not mutually exclusive. This means both events can happen at the same time if the dice shows a 4.
In simple words: When you roll a dice, event E is rolling a 4. Event F is rolling an even number. Since 4 is an even number, both events can happen at the same time. So, they are not "mutually exclusive" because they share an outcome.

🎯 Exam Tip: Remember that two events are mutually exclusive only if they cannot occur at the same time, which means their intersection contains no common elements.

Question 4. Two dice are thrown, then
(i) What is the sample space for pairs?
(ii) What is the sample space for the sum of digits appearing as 8?
Answer:
(i) When two dice are thrown, the sample space for getting a pair (both dice showing the same number) includes:
\( \{(1,1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)\} \)
(ii) The sample space for the sum of the digits on the two dice appearing as 8 includes the following pairs. There are \( 6 \times 6 = 36 \) total outcomes when throwing two dice, representing all possible combinations.
\( \{(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)\} \)
In simple words: When you roll two dice, part (i) asks for all times both dice show the same number, like two 1s or two 6s. Part (ii) asks for all times the numbers on the two dice add up to exactly 8.

🎯 Exam Tip: Always list all possible outcomes systematically for dice problems to avoid missing combinations. For two dice, remember the order can matter, e.g., (2,3) is different from (3,2) unless the problem specifies otherwise.