The mathematics of the coin toss

September 9, 2017 September 2nd, 2019 General Studies

Unit of measurement 4.1

Undertaking 1

In order to guarantee that the pupils understand the basic constructs, the first activity would affect the tossing of merely one coin, as follows:

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Activity 1

Take one coin and flip it. Did you flip a caput or a tail?

Toss the coin once more. What did you flip this clip?

Do you believe that the flip you made the first clip affected the 2nd one you made?

Toss the coin another 20 times, and record for each one whether it was a caput ( H ) or tail ( T ) .

After the activity, the undermentioned points would be discussed:

  • What do you detect about your consequences?
  • Did you flip approximately the same sum of caputs and dress suits?
  • Does there look to be any form in the consequences?

This treatment leads onto the account of the cardinal larning results of this activity. The cardinal vocabulary to be used is highlighted in bold:

Each clip the coin is tossed this is known as anevent– something that has happened.

Theresultof one flip does non impact theresultof the following flip. These two consequences areindependentof each other.

If the result of one event does non impact the result of another event so these events are independent of each other.

As there are two possibleresultsto fliping a coin – caputs or dress suits – and there is an every bit likely opportunity of fliping both, so thechanceof fliping a caput is ? and thechanceof fliping a tail is ? . When you flip a coin several times, about half will come out caputs, and half will come out dress suits.

This activity meets the following learning result from the Adult Numeracy Core Curriculum [ HD2, L2.1 ] :

understand that events are independent when the result of one does non act upon the result of another, e.g. the gender of a babe does non act upon the gender of a 2nd 1

The 2nd activity would so develop this apprehension of single events, and travel onto the topic of combined events. In order for the basic rules to be grasped, this would still remain reasonably simple, and utilize merely two coins.

Activity 2

Take two coins and flip them both. What was theresult?

Toss the coins 20 more times and enter your consequences below:

Number of the flip

First coin

Second coin

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

This first portion of the activity would take onto the undermentioned points of treatment:

  • Make you still agree that the result of the first flip does non impact the result of your 2nd one?
  • When both coins are tossed together, one possible result is two caputs ( HH ) . What are the other possible results?
  • How many possible results are at that place wholly when two coins are tossed together?
  • Do you believe that each result is every bit likely?

The 2nd portion of the activity would so travel onto being able to show this information in the signifier of a tabular array:

Can you utilize this information to make full in thetabular arraybelow, one has already been done to assist:

Result of first flip

Heads

Dress suits

Heads

H, H

Dress suits

This 2nd activity so leads onto the account of the cardinal acquisition results. The cardinal vocabulary to be used is highlighted in bold:

We can see from the tabular array that there are four possibleresultsfrom two coins being tossed together. They all appear one time in the tabular array, so they are all every bit likely. They all depend on what you toss on both coins.

When anresultdepends on theseparate resultsof twosingle events, theeventsare said to becombined. Onesingle eventis fliping a caput with one coin ; a separatesingle eventis fliping a caput with a different coin ; the two events arecombinedto organize an result of two caputs.

In the tabular array, if there are four possibleresultsfor fliping two coins, and one of these is caputs, so there is a 1 in four opportunity of fliping two caputs. So thepossibilityof fliping two coins and them both being caputs is ? .

From all the information gathered so far, the category should so be able to discourse the followers:

  • What is thepossibilityof fliping two dress suits?
  • What is thepossibilityof fliping one caput and one tail?
  • Be careful with the diction in the last inquiry – there is a difference between:
    • What is thepossibilityof fliping one caput and one tail?

and

  • What is thepossibilityof fliping a caput with the first coin and a tail with the 2nd coin?

The last portion of this activity introduces tree diagrams. This could be performed before the treatment above, depending upon how good the group is hold oning the thoughts.

Alternatively of a tabular array, we can besides set this information into a tree diagram like the one below:

H, H = 1/2

T, T = 1/2

This shows that there are two possibilities when the first coin is tossed.

Can you finish the tree diagram for fliping two coins?

H, H =

These activities meet the last two larning results from the Adult Numeracy Core Curriculum [ HD2, L2.1 ] :

  • understand that events are combined when the result depends on the separate result of each independent event, e.g. the likeliness that twins will both be misss
  • record the scope of possible results of combined events in tree diagrams or in tabular arraies

The last activity is a opportunity for consolidation of the cognition and accomplishments introduced.

Activity 3

Toss 3 coins several times and enter your results. When you are confident you are able to, build a tabular array to demo all the possible results of fliping three coins. Remember that each coin flip is an single event, but that these are combined to happen the possible result of fliping three coins.

Construct a tree diagram to demo all the possible results of fliping three coins.

From either your tree diagram or the tabular array, reply the undermentioned inquiries:

How many different results are at that place for fliping three coins?

If you toss a caput on the first spell, does it do it more likely that you will flip a tail with the following coin?

What is the possibility of fliping three caputs?

What is the possibility of fliping two caputs and a tail, in any order?

What is the possibility of fliping a caput with the first coin and dress suits with the following two coins?

How many ways are at that place to flip a tail and two caputs in any order?

Finally, this activity would take onto the undermentioned points of treatment, and these in bend could take onto treatment of how we would cover with cases where the sum of possibilities is excessively big for the methods shown:

  • Do you believe you would be able to enter and utilize your consequences in this manner to happen the possible results for fliping 5 coins?
  • What about 20?
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