If two squares are chosen at random on a chess board, the probability that they have a side in common is

Two squares out of 64 can be selected in ${^{64}C_{2}} = \frac{64 \times 63}{2} = 32 \times 63$ ways The number of ways of selecting those pairs which have a side in common $ = \left( \frac{1}{2} \right)$ $(4 \times 2 + 24 \times 3 + 36 \times 4) = 112$ [Since each of the corner squares has two neighbours each of 24 squares in border other than corner ones has three neighbours and each of the remaining 36 squares have four neighours and in this computation, each pair of squares has been counted twice]. Hence required probability $ = \frac{112}{32 \times 63} = \frac{1}{18}$

If two squares are chosen at random on a chess boa

Notes on Probability

Concepts Used:

Probability

Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.

The definitions of some important terms related to probability are given below:

Sample space

The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.

Sample point

In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.

Experiment

When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.

Event

An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.

Outcome

A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.