A bag contains eleven equally sized marbles, which are numbered. Two marbles are chosen at random and replaced after each selection.
What is the probability that the first marble chosen is shaded and the second marble chosen is labeled with an odd number?
$\frac{10}{121}$
$\frac{24}{121}$
$\frac{6}{11}$
$\frac{10}{11}$
Answer (1)
The probability of the first marble being shaded is 11 S , where S is the number of shaded marbles.
The probability of the second marble being odd is 11 O , where O is the number of odd-numbered marbles.
The probability of both events is 121 S × O .
Assuming S = 1 and O = 10 , the probability is 121 10 .
Explanation
Understand the problem and provided data We are given a bag containing 11 equally sized marbles, each numbered. Two marbles are chosen at random, with replacement after each selection. We want to find the probability that the first marble chosen is shaded and the second marble chosen is labeled with an odd number. However, the number of shaded marbles and odd-numbered marbles are not given. Let's assume that there are S shaded marbles and O odd-numbered marbles.
Calculate individual probabilities The probability of choosing a shaded marble first is the number of shaded marbles divided by the total number of marbles, which is 11 S . Since the marble is replaced, the probability of choosing an odd-numbered marble second is the number of odd-numbered marbles divided by the total number of marbles, which is 11 O .
Calculate the combined probability The probability of both events occurring is the product of the individual probabilities: P ( shaded and odd ) = P ( shaded ) × P ( odd ) = 11 S × 11 O = 121 S × O
We are given the possible answers: 121 10 , 121 24 , 11 6 , 11 10 .
If the probability is 121 10 , then S × O = 10 . Possible values for S and O are (1, 10) or (2, 5) or (5, 2) or (10, 1). If the probability is 121 24 , then S × O = 24 . Possible values for S and O are (3, 8) or (4, 6) or (6, 4) or (8, 3). If the probability is 11 6 = 121 66 , then S × O = 66 . This is not possible since the maximum value for S and O is 11, and 11 × 11 = 121 .
If the probability is 11 10 = 121 110 , then S × O = 110 . This is not possible since the maximum value for S and O is 11, and 11 × 11 = 121 .
Analyze possible scenarios and conclude Without additional information, we cannot determine the exact values of S and O . However, if we assume that S = 2 and O = 5 , then the probability is 121 2 × 5 = 121 10 . If we assume that S = 3 and O = 8 , then the probability is 121 3 × 8 = 121 24 .
Since 121 10 and 121 24 are among the possible answers, we can't determine the correct answer without more information. However, if we assume that the number of shaded marbles is 5 and the number of odd-numbered marbles is 5, then the probability is 121 5 × 5 = 121 25 , which is not among the possible answers. If we assume that the number of shaded marbles is 1 and the number of odd-numbered marbles is 10, then the probability is 121 1 × 10 = 121 10 .
Examples
Consider a game where you draw marbles from a bag. Some marbles are marked as 'special' (shaded), and others have odd numbers. This calculation helps determine your chances of drawing a 'special' marble followed by an odd-numbered one, which could be important for winning the game.
