Arranging Probability Events in Ascending Order

This question asks us to arrange several events based on their probability, from the least probable event to the most probable event. The probability of an event happening is a measure of how likely it is. In this scenario, the events involve hitting a target a certain number of times out of a total number of shots. The probability can be calculated as the ratio of the number of successful hits to the total number of shots.

Calculating Probabilities for Each Event

Let's calculate the probability for each event described:

• Event A: Target is hit 2 times in 20 shots.
  The probability of Event A, denoted as \(P(A)\), is: \[P(A) = \frac{\text{Number of hits}}{\text{Total shots}} = \frac{2}{20}\] Calculating the decimal value: \[P(A) = \frac{2}{20} = \frac{1}{10} = 0.1\]
• Event B: Target is hit 175 times in 200 shots.
  The probability of Event B, \(P(B)\), is: \[P(B) = \frac{175}{200}\] Calculating the decimal value: \[P(B) = \frac{175}{200} = \frac{7 \times 25}{8 \times 25} = \frac{7}{8} = 0.875\]
• Event C: Target is hit 92 times in 100 shots.
  The probability of Event C, \(P(C)\), is: \[P(C) = \frac{92}{100}\] Calculating the decimal value: \[P(C) = \frac{92}{100} = 0.92\]
• Event D: Target is hit 2 times in 5 shots.
  The probability of Event D, \(P(D)\), is: \[P(D) = \frac{2}{5}\] Calculating the decimal value: \[P(D) = \frac{2}{5} = 0.4\]
• Event E: Target is hit 5 times in 13 shots.
  The probability of Event E, \(P(E)\), is: \[P(E) = \frac{5}{13}\] Calculating the decimal value: \[P(E) = \frac{5}{13} \approx 0.3846\]

Comparing Probabilities for Ascending Order

Now we have the decimal probabilities for each event:

• \(P(A) = 0.1\)
• \(P(B) = 0.875\)
• \(P(C) = 0.92\)
• \(P(D) = 0.4\)
• \(P(E) \approx 0.3846\)

To arrange the events in ascending order of their probabilities, we need to order these decimal values from smallest to largest:

Mapping these probabilities back to their respective events, we get the ascending order:

\(P(A)\) < \(P(E)\) < \(P(D)\) < \(P(B)\) < \(P(C)\)

Ascending Order of Events

Based on our calculation, the events arranged in ascending order of their probabilities are:

A, E, D, B, C

Analyzing the Given Options

The question asks to choose the correct answer from the given options. Option (a) presents the order E, A, D, B, C.

Let's list the probabilities for the events in the order given by option (a):

• \(P(E) \approx 0.3846\)
• \(P(A) = 0.1\)
• \(P(D) = 0.4\)
• \(P(B) = 0.875\)
• \(P(C) = 0.92\)

The sequence of probabilities for the order E, A, D, B, C is approximately: 0.3846, 0.1, 0.4, 0.875, 0.92.

Probability Calculation Revision

Comparing the decimal values: 0.1 < 0.3846 < 0.4 < 0.875 < 0.92.

The ascending order of events by probability is A, E, D, B, C.

Additional Information on Probability Concepts

Probability is a fundamental concept in mathematics used to quantify the likelihood of various outcomes. It is always a value between 0 and 1, inclusive. A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event.

When dealing with simple events like hitting a target, the probability is often calculated as the ratio of successful attempts to the total number of attempts. For example, if you flip a fair coin, the probability of getting heads is 1 (favorable outcome) out of 2 (total outcomes), which is \(1/2\) or 0.5.

Comparing probabilities requires expressing them in a comparable format, such as decimals or fractions with a common denominator. Converting fractions to decimals is a common method for easy comparison, as demonstrated in this problem.