Question

The following questions are from ARTIST (reproduced with permission) A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangos. a. Imagine you stick your hand in this refrigerator and pull out a piece of fruit at random. What is the probability that you will pull out a pear? b. Imagine now that you put your hand in the refrigerator and pull out a piece of fruit. You decide you do not want to eat that fruit so you put it back into the refrigerator and pull out another piece of fruit. What is the probability that the first piece of fruit you pull out is a banana and the second piece you pull out is an apple? c. What is the probability that you stick your hand in the refrigerator one time and pull out a mango or an orange?

Short answer

a: \(\frac{3}{44} \), b: \(\frac{15}{484} \), c: \(\frac{7}{44} \).

Step-by-step solution

Calculate Total Number of Fruits

Add up all the fruits in the refrigerator: 6 apples + 5 oranges + 10 bananas + 3 pears + 7 peaches + 11 plums + 2 mangos = 44 fruits total.

Calculate Probability of Pulling a Pear

The probability of pulling out a pear is the number of pears divided by the total number of fruits. There are 3 pears out of 44 fruits: \( P( ext{pear}) = \frac{3}{44} \approx 0.068 \).

Calculate Probability for First Pull as Banana

For the first piece of fruit, calculate the probability of pulling a banana. There are 10 bananas out of 44 fruits: \( P( ext{banana first}) = \frac{10}{44} = \frac{5}{22} \).

Calculate Probability for Second Pull as Apple

Since the first fruit is returned, calculate the probability of pulling an apple next. There are 6 apples out of 44 fruits: \( P( ext{apple second}) = \frac{6}{44} = \frac{3}{22} \).

Calculate Combined Probability for Banana and Apple

Multiply the probability of pulling a banana first and an apple second: \( P( ext{banana and then apple}) = \frac{5}{22} \times \frac{3}{22} = \frac{15}{484} \approx 0.031 \).

Calculate Probability of Mango or Orange

Calculate the probability of pulling out a mango or an orange. There are 2 mangos and 5 oranges, so 7 options out of 44: \( P( ext{mango or orange}) = \frac{7}{44} \approx 0.159 \).

Key concepts

Random Selection

In probability theory, the concept of random selection is pivotal to understanding how likely an event is to occur. When we say a selection is random, it means that every item in the group has an equal chance of being chosen. Imagine reaching into a refrigerator full of various fruits without looking, and pulling one out. That's a random selection because you’re not biased or pre-determined in your choice. To calculate probabilities in such scenarios, you simply compare the number of favorable outcomes to the total number of possible outcomes. Using the fruit example, if you want to know the chance of picking a specific type of fruit, such as a pear, you divide the number of pears by the total number of fruits in the refrigerator. This method embodies the principle of random selection, where each fruit is equally likely to be picked.

Fruit Probability

Fruit probability delves into calculating the likelihood of picking a particular fruit from a collection. Understanding this helps to predict the chance of certain outcomes. For example, if you want to know the probability of pulling an apple, the calculation involves straightforward steps:

• Count the number of apples (in this case, 6).
• Add up the total number of fruits (44 in total).
• The probability is then found by dividing the number of apples by the total fruits: \( \frac{6}{44} \).

In a situation where you withdraw a fruit and put it back before drawing another, each pull remains independent. For instance, the probability of first pulling a banana and then an apple involves multiplying the individual probabilities of each draw, because returning the fruit resets the probabilities: \( \frac{10}{44} \times \frac{6}{44} \). This follows the rule of multiplication for independent events in probability.

Basic Statistics

Basic statistics provides a foundation for understanding data in real-world scenarios. At its core, it involves counting, measuring, and interpreting data to arrive at meaningful conclusions. When you encounter a problem like fruit selection, basic statistical principles guide the calculation of probabilities. For any single fruit, say a mango or an orange, you sum the quantities and divide by the total. This gives you insight into the likelihood of choosing either fruit randomly. The calculation, \( \frac{7}{44} \), reveals the probability of selecting a mango or an orange if they're both just as likely to be picked in a single attempt.Through statistics, you gain a systematic approach to predicting events, measuring likelihoods with fractions and decimals, and understanding inherent randomness in various scenarios, such as our fruit selection problem.