Probability of Choosing 2 Quarters in a Row from a Box of Coins

In this article, we will explore the concept of probability by calculating the likelihood of drawing two quarters consecutively from a box containing 3 dimes, 10 quarters, and 5 nickels. We will follow a step-by-step approach to derive the final answer, making sure to highlight the methodology used in such probability calculations.

Step-by-Step Calculation of Probability

To calculate the probability of choosing 2 quarters in a row, we will follow these steps:

Step 1: Determine the Total Number of Coins

The box contains 3 dimes, 10 quarters, and 5 nickels. We can sum up these quantities to find the total number of coins in the box.

total_coins  3   10   5  18 coins

Step 2: Calculate the Probability of Choosing the First Quarter

The probability of choosing the first quarter is the number of quarters divided by the total number of coins.

P_first_quarter  10 / 18  5 / 9

Step 3: Calculate the Probability of Choosing the Second Quarter

If the first coin chosen was a quarter, then there are now 9 quarters left and 17 coins in total.

P_second_quarter_given_first  9 / 17

Step 4: Calculate the Combined Probability

To find the probability of both events happening (choosing a quarter first and then another quarter), we multiply the probabilities.

P_two_quarters  P_first_quarter * P_second_quarter_given_first  (5 / 9) * (9 / 17)

Calculating this gives:

P_two_quarters  (5 * 9) / (9 * 17)  5 / 17

Final Answer

The probability of choosing 2 quarters in a row is 5 / 17.

Understanding the Probability Calculation

It is important to understand that each choice is dependent on the previous one. For the first draw, the probability of choosing a quarter from a box of 18 coins is 10/18 (which simplifies to 5/9). After drawing one quarter, there are now 17 coins left, of which 9 are quarters. Therefore, the probability of the second coin being a quarter is 9/17.

The overall probability of both events is calculated by multiplying the individual probabilities:

10/18 * 9/17 5/17

This gives an approximate probability of 0.29411.