4. Probability

Counting

4. Probability

Counting

2:24 minutes

Problem 4.4.24a

Textbook Question

ATM You want to obtain cash by using an ATM, but it’s dark and you can’t see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first.

a. What is the probability of selecting a random position and inserting the card with the result that the card is inserted correctly?

Verified step by step guidance

Step 1: Understand the problem. The card can be inserted in four possible orientations: (1) front side up with the beginning of the name entering first, (2) front side up with the end of the name entering first, (3) back side up with the beginning of the name entering first, and (4) back side up with the end of the name entering first.

Step 2: Identify the favorable outcome. The card is inserted correctly only if it is front side up and the beginning of the name enters first. This is one specific orientation out of the four possible orientations.

Step 3: Calculate the total number of possible outcomes. Since there are four possible orientations for inserting the card, the total number of outcomes is 4.

Step 4: Calculate the probability of the favorable outcome. The probability is given by the formula: \( P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Substitute the values: \( P = \frac{1}{4} \).

Step 5: Conclude that the probability of inserting the card correctly is \( \frac{1}{4} \), which represents one favorable outcome out of four possible orientations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it refers to the chance of inserting the ATM card correctly based on the possible orientations of the card. Understanding how to calculate probability involves recognizing the total number of favorable outcomes over the total number of possible outcomes.

Random Selection

Random selection refers to choosing an item or position without any bias or predetermined criteria. In the ATM scenario, it implies that the position where the card is inserted is chosen randomly, which affects the probability of the card being inserted correctly. This concept is crucial for determining how many ways the card can be inserted and how many of those ways are correct.

Outcomes

Outcomes are the possible results of a random experiment. In this case, the outcomes are the different ways the ATM card can be inserted, which include both correct and incorrect orientations. Identifying all possible outcomes is essential for calculating the probability of a successful insertion, as it helps in determining the ratio of favorable outcomes to total outcomes.

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