CLASS 9TH DAY 5

INTRODUCTION

Probability

In everyday life, we come across statements such as

Dad says it will probably rain today, let’s make tea.
I doubt that he will pass the test.
If I get a 6 on the die, I will win the game of Ludo
There is a 50-50 chance of India winning a toss in today’s match.

Probability means possibility. Technically Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it.

Probability can range in between 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.

Activity: Take any coin, toss it ten times and note down the number of times a head and a tail comes up. Record your observations in the form of the following table.

Number of times the coin is tossed	Number of times head comes up	Number of times tails comes up
10

Write down the values of the following fractions

Number of times head comes up/Total number of times coin is tossed

And

Number of times tails comes up/Total number of times coin is tossed

You will find that as the number of tosses gets larger, the values of the fractions come closer to 0.5!! Let n be the total number of trials. The empirical probability P(E) of an event (E) happening, is given by

P(E) = Number of trials in which the event happened/The total number of trials

Now let us consider some other examples.

Example 1 : A coin is tossed 1000 times with the following frequencies: Head : 455,Tail : 545.Compute the probability for each event.

Solution : Since the coin is tossed 1000 times, the total number of trials is 1000. Let’s Call the events of getting a head and of getting a tail as E and F, respectively. Then, the number of times E happens, i.e., the number of times a head comes up, is 455.

So,the probability of E = Number of heads/Total number of trials

i.e.,P (E) = 455/1000 = 0.455

Similarly, the probability of the event of getting a tail F = Number of tails/Total number of trials

i.e.,P(F) =545/1000 = 0.545

Note that in the example above, P(E) + P(F) = 0.455 + 0.545 = 1, and E and F are the only two possible outcomes of each trial.

Example 2 : A die is thrown 1000 times with the frequencies for the outcomes 1, 2, 3,4, 5 and 6 as given in the following table

Outcome	1	2	3	4	5	6
Frequency	179	150	157	149	175	190

Find the probability of getting each outcome.

Solution : Let Ei denote the event of getting the outcome i, where i = 1, 2, 3, 4, 5, 6.Then Probability of the outcome 1 =P(E1)=Frequency of 1/Total number of times the die is thrown

= 179/1000=0.179

Similarly,P(E2) =150/1000 = 0.15,

P(E3) = 157/1000 = 0.157,

P(E4) =149/1000 = 0.149,

P(E5) = 175/1000 = 0.175

And P(E6) =190/1000 = 0.19

Note that P(E1) + P(E2) + P(E3) + P(E4) + P(E5) + P(E6) = 1

Also note that:

(i)The probability of each event lies between 0 and 1.

(ii)The sum of all the probabilities is 1.

(iii)E1, E2, . . ., E6 covers all the possible outcomes of a trial.

Example 3 : There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow?

Solution : The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 2/6 = 1/3.

Example 4 : Fifty seeds were selected at random from each of 5 bags of seeds, and were kept under standardised conditions favourable to germination. After 20 days, the number of seeds which had germinated in each collection were counted and recorded as follows-

Bag	1	2	3	4	5
No. of seeds germinated	40	48	42	39	41

What is the probability of germination of

(i)more than 40 seeds in a bag?

(ii)49 seeds in a bag?

(iii)more than 35 seeds in a bag?

Solution : Total number of bags is 5.

(i) Number of bags in which more than 40 seeds germinated out of 50 seeds is 3.

P(germination of more than 40 seeds in a bag) = 3/5 = 0.6

ii)Number of bags in which 49 seeds germinated = 0.

P(germination of 49 seeds in a bag) = 0/5 = 0

(iii)Number of bags in which more than 35 seeds germinated = 5.So, the required probability = 5/5 = 1

In all the above examples you can observe that Probability can range in between 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.

QUIZ TIME: Lets take a quiz to test your understanding.👇 👇

FUN ACTIVITY: Sixteen Soldiers

BOARD

An expanded Alquerque board is used. Two triangle boards are attached to two opposite sides of an Alquerque board. Each player has 16 pieces that are distinguishable from the other player. Pieces are placed on the intersections (or “points”) of the board, specifically on their half of the Alquerque board, and the nearest triangular board.

RULES

The following rules are based upon Parker’s description:

Players alternate their turns
A player may only use one of their pieces in a turn, and must either make a move or perform a capture but not both.
A piece may move onto any vacant adjacent point along a line.
A piece may capture an opposing piece by the short leap as in draughts or Alquerque. The piece must be adjacent to the opposing piece, and leap over it onto a vacant point immediately beyond. The leap must be in a straight line and follow the pattern on the board. Captures are not mandatory. A piece can continue to capture within the same turn, and may stop capturing any time. The captured piece (or pieces) is removed from the board.
The player who captures all of the other player’s pieces wins.

If you need any help to understand the game follow the link below:

FUN ACTIVITY (REVERSI)

Each of the disks’ two sides corresponds to one player; they are referred to here as light and dark after the sides of Othello pieces, but any counters with distinctive faces are suitable. The game may for example be played with a chessboard and Scrabble pieces, with one player letters and the other backs.

The historical version of Reversi starts with an empty board, and the first two moves made by each player are in the four central squares of the board. The players place their disks alternately with their colors facing up and no captures are made. A player may choose to not play both pieces on the same diagonal, different from the standard Othello opening. It is also possible to play variants of Reversi and Othello where the second player’s second move may or must flip one of the opposite-colored disks (as variants closest to the normal games).

For the specific game of Othello (differing from the historical Reversi), the rules state that the game begins with four disks placed in a square in the middle of the grid, two facing white side up, two pieces with the dark side up, with same-colored disks on a diagonal with each other.

Convention has initial board position such that the disks with dark side up are to the north-east and south-west (from both players’ perspectives), though this is only marginally meaningful to play (where opening memorization is an issue, some players may benefit from consistency on this). If the disks with dark side up are to the north-west and south-east, the board may be rotated by 90° clockwise or counterclockwise. The dark player moves first.