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A tree diagram is a valuable tool in probability that helps visualize and solve problems by organizing information into a branching structure. It allows us to see all possible outcomes and their probabilities. In the context of our survey on smoking habits among different occupational classes, the tree diagram has two levels:

• First level: Represents the different occupational classes and their proportions in the population—managerial and professional (43%), intermediate (34%), and routine and manual (23%).
• Second level: Denotes the probability of smoking within each occupational group—20%, 29%, and 38% respectively.

By mapping these probabilities, the tree diagram simplifies the complex problem of calculating the overall percentage of smokers. Each path down the tree represents a combined probability, making it a powerful visualization tool in understanding statistical data.

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In our exercise, we calculate the smoking rate within each occupational class, given that a person belongs to that class. This is a perfect example of conditional probability.To compute the probability that a randomly selected man smokes and belongs, for instance, to the managerial and professional group, we multiply the probability of being in that occupation (43%) by the probability of being a smoker within that occupation (20%).Formally, if \( A \) is the event of being in a specific class and \( B \) the event of smoking, the conditional probability is calculated as:\[ P(B|A) = P(A \cap B) / P(A) \]Using this for each group helps us find the percentage of smokers in each occupational class.

Percentages are a fundamental aspect of probability and statistics, providing a way to express proportions as parts of a hundred. In surveys and data like the one we're examining, they make comparisons straightforward and intuitive. In our smoking survey:

• The occupational proportions are given as percentages of the total population — for example, the managerial and professional group composes 43% of all men.
• The smoking rate within each group is also given as a percentage—20%, 29%, and 38% respectively for the three classes.

By expressing these figures as percentages, we can easily visualize and understand the proportion of smokers within each occupational class and across the entire population. This makes calculating the overall number of smokers and identifying trends much more accessible.

Survey statistics involve collecting, analyzing, and interpreting data from a sample population to gain insights about a larger population. This exercise is a great illustration of how survey statistics can be used to understand social behaviors such as smoking. Key elements include:

• Identifying the sample groups—in our case, occupational classes.
• Measuring specific behaviors—here, the prevalence of smoking within each class.
• Using statistical tools like tree diagrams and conditional probabilities to infer the data.

The survey conducted provides us with crucial data points, such as the percentage of smokers in each class and overall trends. Analyzing this data through survey statistics allows us to draw meaningful conclusions about social patterns, like how smoking rates differ by occupation in the British population.