One ticket will be drawn at random from each of the two...

One ticket will be drawn at random from each of the two boxes shown below: $( \mathrm { A } ) \lfloor [ 1 ] [ 2 ] [ 3 ]$ $( \mathrm { B } ) \lfloor [ 1 ] [ 2 ] [ 3 ] [ 4 ]$ Find the chance that: (a) The number drawn from A is larger than the one from B. (b) The number drawn from A equals the one from B. (c) The number drawn from A is smaller than the one from B.

One ticket will be drawn at random from each of the two boxes shown below:

$( \mathrm { A } ) \lfloor [ 1 ] [ 2 ] [ 3 ]$ $( \mathrm { B } ) \lfloor [ 1 ] [ 2 ] [ 3 ] [ 4 ]$

Find the chance that:
(a) The number drawn from A is larger than the one from B.
(b) The number drawn from A equals the one from B.
(c) The number drawn from A is smaller than the one from B.

Key Concepts

Counting Outcomes

Counting techniques are used to determine the size of the sample space by considering the number of choices available from each independent event. Here, since one ticket is drawn from each box and the drawing from each box is independent, the total number of outcomes is the product of the number of tickets in each box.

Probability Calculation

Probability measures the likelihood of an event occurring and is defined as the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. This concept applies directly in comparing the drawn numbers by computing the ratios for events where one number is larger than, equal to, or smaller than the other.

Independent Events

The concept of independent events means that the outcome of one event (drawing a ticket from the first box) does not affect the outcome of the other event (drawing a ticket from the second box). This property allows for the multiplication of the number of outcomes from each independent event to determine the size of the overall sample space.

Comparison of Random Variables

Comparing two random variables involves evaluating relationships such as greater than, equal to, or less than. In this problem, the two drawn numbers represent random variables, and determining the probability of one being greater than, equal to, or less than the other involves systematically comparing all possible pairs of outcomes.

Sample Space

The sample space represents the set of all possible outcomes from a probabilistic experiment. In this problem, each outcome corresponds to a pair of numbers drawn from the two boxes. Understanding the sample space is crucial because it defines the total number of equally likely outcomes over which probabilities are calculated.

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