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A bag contains four red marbles, three yellow marbles, and three purple marbles. You will randomly select one marble from the bag.



Which statements are true regarding the scenario? Check all that apply.

  • The probability you select a red marble is $\frac{3}{10}$.
  • The probability you select a purple marble is $\frac{3}{10}$.
  • $P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 1$
  • The probability you select a green marble is 0.
  • The probability you select a yellow marble is 3.

Verified Answer :

  • Calculate the total number of marbles: \(4 + 3 + 3 = 10\).
  • Determine the individual probabilities: \(P(\text{Red}) = \frac{4}{10}\), \(P(\text{Yellow}) = \frac{3}{10}\), \(P(\text{Purple}) = \frac{3}{10}\).
  • Validate the statements based on these calculations:
  • The probability you select a purple marble is \(\frac{3}{10}\) (\(\boxed{\text{True}}\)).
  • \(P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 1\) (\(\boxed{\text{True}}\)).
  • The probability you select a green marble is \(0\) (\(\boxed{\text{True}}\)).

Explanation

  1. Analyze the bag composition and calculate individual probabilities

We start by analyzing the composition of the bag. The bag contains:

  • 4 red marbles
  • 3 yellow marbles
  • 3 purple marbles

The total number of marbles in the bag is \(4 + 3 + 3 = 10\). The probability of selecting a marble of a specific color is determined by the ratio of the number of marbles of that color to the total number of marbles. Therefore, we calculate the individual probabilities as follows:

  • Probability of Red (\(P(\text{Red})\)) = \(\frac{4}{10}\)
  • Probability of Yellow (\(P(\text{Yellow})\)) = \(\frac{3}{10}\)
  • Probability of Purple (\(P(\text{Purple})\)) = \(\frac{3}{10}\)
  1. Evaluate the truth value of each statement

Now we evaluate each of the given statements:

  1. "The probability you select a red marble is \(\frac{3}{10}\)": Since \(P(\text{Red}) = \frac{4}{10}\), this statement is False.
  2. "The probability you select a purple marble is \(\frac{3}{10}\)": Since \(P(\text{Purple}) = \frac{3}{10}\), this statement is True.
  3. "\(P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 1\)": The sum is \(\frac{4}{10} + \frac{3}{10} + \frac{3}{10} = \frac{10}{10} = 1\), so this statement is True.
  4. "The probability you select a green marble is \(0\)": Since there are no green marbles in the bag, the probability is indeed \(0\). This statement is True.
  5. "The probability you select a yellow marble is \(3\)": Probabilities must be between \(0\) and \(1\). The probability is \(\frac{3}{10}\), not \(3\), so this statement is False.

Examples

This type of probability problem is foundational for understanding how to calculate likelihoods in real-world scenarios. For example, if you are managing a quality control process where a bin contains different types of items, calculating these simple probabilities helps you predict how likely it is to pull a specific type of item at random, ensuring you understand the distribution of your inventory.

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