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Understanding Counting and Probability: Tossing Coins and Rolling Dice

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deanna
12 diapositives · 329 Views · Jul 27, 2014

This resource delves into basic concepts of counting and probability through practical examples such as tossing two coins and rolling a pair of dice. It explains the outcomes for heads or tails when two coins are tossed, detailing probabilities based on all equally likely outcomes. Additionally, it covers rolling two dice and calculates probabilities for specific outcomes such as sums or even numbers. The document also explores counting consecutive integers, offering real-world applications to deepen understanding of mathematical concepts.

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Understanding Counting and Probability: Tossing Coins and Rolling Dice
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  1. Counting and Probability

  2. Tossing Two Coins Imagine tossing two coins and observing whether 0, 1, or 2 heads are obtained. Below are the results after 50 tosses

  3. Set of possible outcome of tossing two coins Outcomes for 2 heads Outcomes for 1 head Outcomes for 0 heads

  4. In case an experiment has finitely many outcomes and all outcomes are equally likely to occur, the probability of an event (set of outcomes) is just the ratio of the number of outcomes in the event to the total number of outcomes (sample space).

  5. NOTATION For a finite set A, the number of elements of A is denoted N(A) or |A|

  6. Probabilities Tossing Two Coins Outcomes for 2 heads Outcomes for 1 head

  7. Rolling A Pair of Dice A die is one of a pair of dice. Suppose a blue die and a gray die are rolled together, and the numbers of dots that occur face up on each are recorded. The possible outcomes can be listed as follows, where in each case the die on the left is blue and the one on the right is gray.

  8. NOTATION The outcome can be represented as (2,4) or 24 The outcome can be represented as (5,3) or 53 With this notation the set of all possible outcomes can be represented as

  9. What is the probability that the sum of the outcome is 5? What is the probability that outcome is an even numbers?

  10. Counting Consecutive Integers Determine the size of each of the following sets {0, 1, 2, …., n} {3, 4, 5, …., 100} {3+0,3+1,…,3+(100-3)} {37,38,39,….,10278} {37+0,37+1,…,37+(10278-37)}

  11. What is the size of the set of integers {n, n+1, n+2,…., m} ?

  12. Application of Counting a List • How many three-digit integers (integers from 100 to 999 inclusive) are divisible by 5? (Describe that set as a list) b. What is the probability that a randomly chosen three-digit integer is divisible by 5?