Probability Tree Diagram

Probability Tree Diagram 

A probability Tree Diagram  is simply a way of representing a sequence of events.

Tree diagram particularly illustrates the PROBABILITY since they record all the possible outcomes in a clear and uncomplicated way.

To draw the tree diagram:

• The first event is always represented by a dot.
• From the dot, the branches are drawn to represent all possible outcomes. That is, the outcomes of an event is usually written at the end of the branch
• The probability of each branch is written on the branches.
• We have to repeat the process for the number of events asked.

Example

1. Draw a Tree diagram For tossing a coin

Solution

Step 1:

The first event starts by dot.

           

Step 2:

From the dot, the branches are drawn to represent all possible outcomes.

when we toss a coin, the possible outcomes are heads and tails

let , heads = H

       tails = T

so,

Step 3 

here, we are tossing just a coin for once.

So now we can list out the possible outcomes.

Step 4:

Now, we have to write the outcomes probabilities on the corresponding branches.

That is, Probability of getting heads when toss a coin = 1/2

            Probability of getting tails when toss a coin = 1/2

Hence the above diagram represents the complete Tree Diagram when tossing a coin

Example 2 

1. Draw a Tree diagram For two tosses of a coin

Solution

Step 1:

The first event starts by dot.

Step 2:

From the dot, the branches are drawn to represent all possible outcomes.

when we toss a coin, the possible outcomes are heads and tails

let , heads = H

       tails = T

so,

Step 3:

here we are tossing twice, so again we have to list the possible outcomes for the second toss.

Even when we toss again the same possible outcomes we are going to get that is Heads and Tails.

Step 4 :

Now lets list out all the possible outcomes when we toss a coin twice.

 

Step 4 :

Now, we have to write the outcomes probabilities on the corresponding branches.

That is, Probability of getting heads when toss a coin = 1/2

            Probability of getting tails when toss a coin = 1/2

Step 5 :

Now lets write the probabilities of the final outcomes.

The final outcomes, that is sample space S= { (HH) , (HT) , (TH) , (TT) }

The probability of getting HH = 1/4

The probability of getting HT = 1/4

The probability of getting TH = 1/4

The probability of getting TT = 1/4

Probability

Experiment

    A  repeatable procedure with a set of possible results.

Example

 Tossing a coin , rolling a die 

Outcome

    An outcome is a possible result of an experiment.

Example

    Getting a head or a tail when we toss a coin. So head or tail are all outcomes.

    Getting 1 , 2 , 3 , 4 , 5 , 6 are all outcomes when we roll a die.

Event 

One or more possible outcomes of an experiment.

Example

    Getting a tail when tossing a coin (one outcome)

    Getting a 5  when rolling a die (one outcome)

    Getting an odd number when rolling a die. The possible outcomes are 1 , 3 , 5 (more than one outcome) 

Sample Space

    All the possible outcomes of an experiment. It is usually denoted by the letter S. It can be written using { } (set notation).

Example

    1.When tossing a coin, the possible outcomes are head or tail.

so, Sample space (S) = { Head , Tail }

Number of sample space can be written as n(S).

here, n(S) = 2

    2. When rolling a die, the possible outcomes are 1 , 2 , 3 , 4 , 5 , 6.

S = { 1 , 2 , 3 , 4 , 5 , 6 }

n(S) = 6 

Sample point

    Each one of the possible outcome of an event.

Example 

    1. If a die is rolled, the possible outcomes are S= { 1 , 2 , 3 , 4 , 5 , 6 }

here, there are 6 sample points such as 1 , 2 , 3 , 4 , 5 , 6.

    2. When tossing a coin the possible outcomes are S= { Head , Tail }

here, there are 2 sample points such as Head and Tail.

Probability

    Probability is simply how likely something is to happen . 

Probability(Event) = No.of favorable outcomes / Total no.of outcomes 

 Example

    1.When a coin is tossed , there are 2 possible outcomes. Head (H) and Tails (T). Find the probability of getting Head.

We can say that 

 Sample space (S) = { H ,T}

                  n(S) = 2 

let us assume 'A' be an event  getting head

A={H} 

n(A) = 1  [ n(A) means number of  possible event getting head]

Probability(Event) = No.of favorable outcomes / Total no.of outcomes 

that is here, 

Probability (getting head) = n(A) / n(S)     

                                          = 1 / 2

                                           = 0.5

                                  = 50% (to convert into percentage we have to multiply by 100, so 0.5)

Types of Events

     There are 5 types of events. They are 

             1. Certain Event

             2. Likely Event

             3. Equally Likely Event

             4. Unlikely Event

             5. Impossible Event

Certain Event

    An Event is certain , if it will always happen.

Example

A box contains 10 Red balls. Since all the balls are in Red,the outcome will always be a red ball. So picking a red ball from the box is Certain Event.

Likely Event

    An Event is Likely if it has good chances of happening.

Example

A box contains 10 Red balls and 2 Green balls. Since most of the balls are in Red, picking a red ball from the box has good chance of happening. So we can say it as Likely Event.

Equally Likely Event

    An event is equally likely if it has an even chance of happening.

Example

Getting head when tossing a coin

Unlikely Event

    An event is unlikely if it has a poor chance of happening.

Example

A box contains 10 Red balls and 2 Green balls. Since most of the balls are in Red, picking a green ball from the box has poor chance of happening. So we can say it as Unlikely Event.

Impossible Event

    An event is impossible if it will never happen or has no chance of happening.

Example

Getting a number greater than 7 is Impossible event, since the possible outcomes are

 { 1 , 2 , 3 , 4 , 5 , 6 }. 

 

Key Points

• The probability always lies from 0 to 1
• The total probability is always 1
• If Probability = 1, then the event is called as Certain
• If Probability = 0, then the event is called as Impossible
• If Probability = 1/2 = 0.5, then the event is called as Equally Likely
• If Probability lies between 0 and 1/2 (0 and 0.5), then the event is called as  Unlikely event
•  If Probability lies between 1/2 and 1, then the event is called as  Likely event