Question

Four numbers are multiplied together. Then the probability that the product will be divisible by 5 or 10 is : (a) \(\frac{169}{625}\) (b) \(\frac{369}{625}\) (c) \(\frac{169}{1626}\) (d) none of these

Short answer

Answer: (b) \(\frac{369}{625}\)

Step-by-step solution

Understand possible outcomes

There are two possible outcomes when choosing a number: it can be either divisible by 5 (in this case, we consider it a success) or not divisible by 5 (in this case, it is a failure). When four numbers are multiplied together, there are \(2^4 = 16\) possible combinations of successes and failures. Out of these 16 possibilities, at least one number must be divisible by 5 for the product to be divisible by 5.

Determine the probability of success and failure

In order for the product to be divisible by 5, one or more of the four numbers must be divisible by 5. To determine the probability of success (choosing a number divisible by 5), we will first find the probability of failure (choosing a number not divisible by 5). Since there are 5 possible values for the last digit (0, 1, 2, 3, and 4), there is a \(\frac{1}{5}\) chance of choosing a number divisible by 5 (success) and a \(\frac{4}{5}\) chance of choosing a number not divisible by 5 (failure).

Calculate the probability of getting at least one success

To find the probability of getting at least one success (number divisible by 5) in four attempts, we can calculate the probability of getting no successes (all failures) and subtract it from 1 (total probability). The probability of getting no successes (all failures) in four attempts is: \(P(\text{no success}) = (\frac{4}{5})^4\) Now, we will subtract the probability of no success from the total probability to get the probability of getting at least one success: \(P(\text{at least one success}) = 1 - P(\text{no success}) = 1 - (\frac{4}{5})^4\)

Calculate the final probability

Now, we will evaluate the expression found in Step 3: \(P(\text{at least one success}) = 1 - (\frac{4}{5})^4 = 1 - \frac{256}{625} = \frac{369}{625}\)

Choose the correct answer

Looking at the options given, we can see that the calculated probability is in the options. Therefore, the correct answer is: (b) \(\frac{369}{625}\)

Key concepts

Probability Theory

At its core, probability theory is a branch of mathematics that deals with the likelihood of different outcomes. It's a way to quantify uncertainty and make educated predictions about events, such as rolling dice or drawing cards.

When we talk about probability, we often refer to it as a number between 0 and 1, where 0 signifies an impossible event and 1 indicates a certainty. For example, if we were flipping a fair coin, the probability of getting heads would be 0.5, since there are two equally likely outcomes.

In the context of our exercise, where we want the product of four numbers to be divisible by 5 or 10, we're looking at a basic probability scenario with two outcomes: 'success' (divisible by 5) and 'failure' (not divisible by 5). We calculate the probability by considering all possible combinations and focussing on the favorable outcomes. This simple yet powerful framework allows us to systematically approach and solve a wide range of problems in different fields, from mathematics to finance, and from engineering to social sciences.

Combinatorial Probability

Combinatorial probability is a part of probability theory that focuses on counting and combining elements to calculate probabilities. It concerns itself with the likelihood of certain combinations or sequences of events occurring.

Let's consider a lottery as a real-world example. The probability of winning is calculated based on the number of possible combinations of numbers drawn; this calculation is grounded in combinatorial probability.

In our textbook exercise, we use combinatorial probability to evaluate all possible combinations of four numbers being multiplied, where each number can either be divisible by 5 (we'll call this a 'hit') or not (a 'miss'). By calculating the chance of all misses, we can deduce the probability of at least one hit, which is necessary for the product to be divisible by 5 or 10. This approach of considering all potential 'misses' and subtracting from the total possible outcomes is a standard technique in combinatorial probability, allowing for a systematic approach to seemingly complex probability problems.

Divisibility Rules

Divisibility rules are simple shortcuts that help us quickly determine if one number is divisible by another without performing the actual division. These rules are fundamental in number theory and are particularly helpful in various mathematical problems, including probability exercises.

For instance, a number is divisible by 5 if its last digit is either 0 or 5. This rule springs from the fact that our base-10 number system is such that the last digit dictates divisibility by 5 and 10. Thanks to this convenient rule, we can swiftly assess the outcomes that will result in our product being a multiple of 5 or 10.

In solving our problem, we apply this divisibility rule to determine successful outcomes much faster than if we were to consider the divisibility of each number individually. Divisibility rules, therefore, not only expedite calculations but they also simplify our understanding of the problem's constraints and enable us to streamline the steps in finding our probability solution.