Probability - Introduction

A die is thrown once. Find the probability of getting (i) a prime number; (ii) a number lying between $2$ and $6$.

The possible outcomes when a die is thrown are $1, 2, 3, 4, 5, 6$. So, total outcomes $n(S) = 6$. (i) Prime numbers are $2, 3, 5$. Number of favorable outcomes $n(E) = 3$. $P(\text{prime number}) = \frac{3}{6} = \frac{1}{2}$ (ii) Numbers between $2$ and $6$ are $3, 4, 5$. Number of favorable outcomes $n(F) = 3$. $P(\text{number between 2 and 6}) = \frac{3}{6} = \frac{1}{2}$

We identify the total number of outcomes first ($6$ for a die). Then we count how many of those outcomes satisfy the given condition to find the probability using the ratio $\frac{n(E)}{n(S)}$.

If $P(E) = 0.07$, what is the probability of 'not $E$'?

We know that for any event $E$, $P(E) + P(\bar{E}) = 1$. Given $P(E) = 0.07$, $P(\bar{E}) = 1 - P(E)$ $P(\bar{E}) = 1 - 0.07$ $P(\bar{E}) = 0.93$

This uses the concept of complementary events, where the sum of the probability of an event happening and the probability of it not happening is always equal to $1$.

One card is drawn from a well-shuffled deck of $52$ cards. Calculate the probability that the card will be a king of red color.

Total number of outcomes $n(S) = 52$. In a deck of $52$ cards, there are $2$ red kings (one of Hearts and one of Diamonds). Number of favorable outcomes $n(E) = 2$. $P(\text{Red King}) = \frac{2}{52} = \frac{1}{26}$

A standard deck has $4$ kings in total, $2$ of which are red and $2$ of which are black. The probability is the number of red kings divided by the total number of cards.