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Probability progression


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Last updated 10 January 2013 11:09 by NZTecAdmin

Many everyday applications of statistics require an understanding of the nature of probability and how mathematical models and relative frequency can be used to describe the likelihood of particular outcomes (events).

Most adults will be able to:
Activities
1.
  • identify all possible outcomes in situations involving simple (single-stage) chance
  • use words to describe the likelihood of particular outcomes (events).

Learners identify all possible outcomes in situations such as tossing a coin and rolling a dice.

Learners use words to position events on a continuum that goes from impossible to certain.

Describing likelihood

Learners develop an understanding of the words used to describe the probability or likelihood of events.

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3.
  • use fractions to express the probability of events
  • recognise uncertainty in simple (singlestage) chance situations.

Learners use fractions to express probabilities in simple chance situations. For example, if a coin is tossed, the probability of it landing as a head is 1/2 .

Learners know that, in such situations, the outcome is not influenced by earlier outcomes. If a coin is tossed three times, and three heads are obtained, this makes it no more or less likely that the result of the next toss will be a head.

Understanding chance

Learners develop their concept of chance by discussing the likelihood of different events and use fractions to assign likelihoods to outcomes.

  • use fractions to express the probability of events
  • recognise uncertainty in simple (singlestage) chance situations.
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4.
  • use relative frequency to provide an estimate of the probability of an event
  • use fractions, ratios and percentages to express probabilities
  • compare the results of trials or observations with expectations based on models.

Learners understand that they can estimate the likelihood of an event by doing repeated trials or observations and then comparing the number of ‘successes’ with the number of trials. For example, the probability that a goal shoot will get her next shot in the net can be estimated by comparing successes with attempts over a number of games.

For simple chance events, learners can interpret probabilities expressed as fractions, percentages and ratios (odds).

Learners understand that, while they can’t determine which of the various possible outcomes will occur next, they may be able to determine mathematically the probabilities of each outcome. For example, the probability of drawing an ace from a standard pack of 52 playing cards is 4/52 = 1/13.

Theoretical probability

Learners develop their understanding of the difference between theoretical and experimental probabilities.

Using frequencies to predict

Learners use the frequencies of outcomes to predict the likelihood that an event will occur, and learn that probabilities are not absolute predictors in the short run.

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5.
  • determine the probabilities in simple multi-stage probability situations
  • apply the law of large numbers to probability situations.

In simple, multi-stage chance situations (where the events are independent), learners can determine probabilities by using, for example, systematic lists or two-way arrays (tables).

Learners can recognise complementary and mutually exclusive events. For example, a learner knows that when two dice are rolled, ‘double’ and ‘not a double’ are both complementary (one or other must occur) and mutually exclusive (they can’t both occur) events.

Learners know there are situations in which probabilities cannot be determined theoretically. In such cases, the relative frequency of an event can be used to estimate its probability.

Learners know that, in determining the relative frequency of an event, the greater the number of trials, the more accurate the estimate (this is the law of large numbers). For example, the accuracy of a particular weather forecaster can’t be determined from a single forecast.

Theoretical probability

Learners develop their understanding of the difference between theoretical and experimental probabilities.

Using frequencies to predict

Learners use the frequencies of outcomes to predict the likelihood that an event will occur, and learn that probabilities are not absolute predictors in the short run.

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6.
  • determine the probabilities in more complex multi-stage chance situations
  • apply the notion of ‘expected value’ to probability situations.

Learners can:

  • understand that the probability of an event can sometimes depend on previous outcomes (such events are said to be dependent), and a learner can apply this understanding to more complex probability situations
  • determine probabilities in multi-stage chance situations. For example, the probability of a first-division Lotto win is 6/40 x 5/39 x 4/38 x 3/37 x 2/36 x 1/35 (about 1 in 4 million). Situations can be modelled using such means as systematic lists or tree diagrams
  • apply the notion of ‘expected value’ to probability situations such as games (with pay-offs) and insurance premiums. For example, if the ‘house advantage’ on a gaming machine is 10 percent, over the long run, punters will get back only 90 percent of what they paid in.

Fair games?

Learners explore issues related to gambling to further develop their concept of chance and to develop an understanding of expected value.

Lotto winners

Learners learn to use tree diagrams, organised lists and how to multiply probabilities to calculate the theoretical probability of winning Lotto.

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