Question

Chandu picks up a letter from the English alphabet and finds it to be a vowel. Find the probability that it is either e or \(\mathrm{i}\). (1) \(\frac{2}{5}\) (2) \(\frac{3}{5}\) (3) \(\frac{1}{5}\) (4) \(\frac{4}{5}\)

Short answer

Answer: (1) \(\frac{2}{5}\).

Step-by-step solution

1. Identifying the event and the sample space

Since we are considering vowels, the sample space consists of 5 vowels: {a, e, i, o, u}. Our event of interest (E) is picking either 'e' or 'i'. So, E = {e, i}.

2. Computing the probability

To calculate the probability of event E occurring, we need to divide the number of favorable outcomes by the total number of outcomes in the sample space. In this case, there are 2 favorable outcomes (e, i), and a total of 5 outcomes in the sample space (a, e, i, o, u). Using the formula for probability, P(E) = \(\frac{\mathrm{Number \ of \ favorable \ outcomes}}{\mathrm{Total \ number \ of \ outcomes}}\).

3. Substituting the values

Substitute the number of favorable outcomes (2) and the total number of outcomes (5) into the formula. P(E) = \(\frac{2}{5}\) Hence, the probability that Chandu picks either 'e' or 'i' is \(\frac{2}{5}\). The correct answer is (1) \(\frac{2}{5}\).

Key concepts

Understanding Sample Space

When we talk about probability in mathematics, the concept of a 'sample space' is fundamental. In essence, the sample space is the set of all possible outcomes for a given experiment or event. It gives us a complete overview on what could potentially happen. For example, when Chandu picks a letter from the English alphabet hoping to find a vowel, the sample space is limited to just the vowels since we are not considering any consonants. Therefore, here our sample space is \( \{a, e, i, o, u\} \), which contains all five possible vowels one could pick.

To make this clearer to students, let's equate the concept of a sample space to picking a colored ball from a bag. If the bag only contains red, blue, and green balls, then the sample space for picking a ball is \( \{red, blue, green\} \). The sample space is crucial as it forms the basis for calculating probabilities, illustrating all scenarios before any action is taken, and therefore setting the boundaries for our expectations.

Identifying Favorable Outcomes

Favorable outcomes are simply the outcomes we are interested in, or specifically, those that are 'favorable' to the event we're observing. They are a subset of the sample space. In Chandu's case, the act of picking a vowel out of the alphabet results in a sample space of five vowels. However, his interest is in picking either an 'e' or an 'i', which makes his favorable outcomes the set \( \{e, i\} \).

Relation to Real-Life Scenarios

In real life, probabilities play a role in decision making, like forecasting weather or planning events. For instance, if the weather forecast predicts a 70% probability of rain, the favorable outcome for someone hoping to avoid getting wet is the remaining 30% chance that it doesn't rain. In Chandu's scenario, since there are 5 vowels, but he is only happy with 2 (either 'e' or 'i'), the favorable outcomes are those 2 vowels he is hoping to pick.

Applying the Probability Formula

Mathematics gives us a way to quantify uncertainty through the probability formula, which is essentially a ratio: \[ P(E) = \frac{\mathrm{Number\ of\ favorable\ outcomes}}{\mathrm{Total\ number\ of\ outcomes}} \]. This formula allows us to compute the probability (P) of an event (E) happening. So, in practice, if we consider Chandu's situation, we substituted the number of favorable outcomes (2) and the total number of outcomes in our sample space (5) into this formula, resulting in \( P(E) = \frac{2}{5} \).

Understanding this formula is core to grasping how probability works. It's also important to remember that probabilities range from 0 to 1, where 0 means the event cannot happen and 1 means it will happen for sure. A probability of \( \frac{2}{5} \) shows there's less than a 50% chance of picking an 'e' or an 'i', but it is more likely than not picking either of these vowels.