Page 1 PROBABILITY The word Probability is commonly used to represent uncertainty of an event. Example (i) Probably, it may rain today. (ii) Probably, he may stand first in the class. So, Probability is used to mention the chances or possibilities of a thing to occur. In other words, it represents the chances/possibilities for an event to occur out of an operation/experiment. Example If we throw a dice, then we may obtain any number out of the six numbers of dice. In this case, throwing the dice is the experiment/operation and getting a particular number is an event. To study the â€˜Probabilityâ€™ mathematically, we will first study some terms related to â€˜Probabilityâ€™. 1. Experiment The word â€˜experimentâ€™ means, the operation which produces the result. The experiments are divided into two categories. (i) Deterministic Experiment Deterministic experiments are those experiments, which on repeating under same conditions produce same result. Example Science or engineering events under same conditions produce same or almost same results. (ii) Probabilistic / Random Experiments Probabilistic or random experiments are those experiments, which on repeating under same conditions do not produce same results. In other words, these experiments produce one single result out of several possible results. Example Throwing a dice under same conditions will produce different results each time or maximum times. 2. Event The result of an experiment is called event. These are following types of events: - (i) Elementary Event The basic result of a random experiment is called elementary event. That means, the final single result obtain by an experiment is called elementary events. Example (i) On tossing a can randomly, we either get head or tail. Then, H = Getting H (head) T = Getting T (tail) So, H and T are elementary events, which are the results of random experiment tossing the coin. (ii) On tossing two coins simultaneous, we get different combinations of head and tail. Then, HT = Getting head (H) on first and tail (T) on second. HH = Getting head (H) on first and head (H) on second TH = Getting tail (T) and first and head (H) on second TT = Getting tail (T) on first and tail (T) on second So, HT, HH, TH and TT are elementary events. (iii) On throwing a dice, we get different numbers Then, - 1 N Getting Number 1 - 2 N Getting number 2 - 3 N Getting number 3 - 4 N Getting number 4 - 5 N Getting number 5 - 6 N Getting number 6 So 5 4 3 2 1 N , N , N , N , N and 6 N are elementary events. (ii) Compound Event The result of a random experiment made up of more than one elementary event is called compound event. Example On throwing a dice, getting a number is elementary event. In such case, getting an event number is a compound event, which is made of three elementary events, i.e. getting number 1, getting number 3 and getting number 5. (iii) Possible Event All the possible results, which may produced by the experiment are called possible events Example (i) On tossing a coin, we may get 2 possible results. H â€“ Getting Head (H) T â€“ Getting tail (T) So, number of possible events is 2, i.e. H and T. (ii) On throwing a dice, we may get 6 possible results. - 1 N Getting number 1 - 2 N Getting number 2 - 3 N Getting number 3 - 4 N Getting number 4 - 5 N Getting number 5 - 6 N Getting number 6. So, number of possible events is 6, i.e. 1 N , 2 N , 3 N , 4 N , 5 N , 6 N Page 2 PROBABILITY The word Probability is commonly used to represent uncertainty of an event. Example (i) Probably, it may rain today. (ii) Probably, he may stand first in the class. So, Probability is used to mention the chances or possibilities of a thing to occur. In other words, it represents the chances/possibilities for an event to occur out of an operation/experiment. Example If we throw a dice, then we may obtain any number out of the six numbers of dice. In this case, throwing the dice is the experiment/operation and getting a particular number is an event. To study the â€˜Probabilityâ€™ mathematically, we will first study some terms related to â€˜Probabilityâ€™. 1. Experiment The word â€˜experimentâ€™ means, the operation which produces the result. The experiments are divided into two categories. (i) Deterministic Experiment Deterministic experiments are those experiments, which on repeating under same conditions produce same result. Example Science or engineering events under same conditions produce same or almost same results. (ii) Probabilistic / Random Experiments Probabilistic or random experiments are those experiments, which on repeating under same conditions do not produce same results. In other words, these experiments produce one single result out of several possible results. Example Throwing a dice under same conditions will produce different results each time or maximum times. 2. Event The result of an experiment is called event. These are following types of events: - (i) Elementary Event The basic result of a random experiment is called elementary event. That means, the final single result obtain by an experiment is called elementary events. Example (i) On tossing a can randomly, we either get head or tail. Then, H = Getting H (head) T = Getting T (tail) So, H and T are elementary events, which are the results of random experiment tossing the coin. (ii) On tossing two coins simultaneous, we get different combinations of head and tail. Then, HT = Getting head (H) on first and tail (T) on second. HH = Getting head (H) on first and head (H) on second TH = Getting tail (T) and first and head (H) on second TT = Getting tail (T) on first and tail (T) on second So, HT, HH, TH and TT are elementary events. (iii) On throwing a dice, we get different numbers Then, - 1 N Getting Number 1 - 2 N Getting number 2 - 3 N Getting number 3 - 4 N Getting number 4 - 5 N Getting number 5 - 6 N Getting number 6 So 5 4 3 2 1 N , N , N , N , N and 6 N are elementary events. (ii) Compound Event The result of a random experiment made up of more than one elementary event is called compound event. Example On throwing a dice, getting a number is elementary event. In such case, getting an event number is a compound event, which is made of three elementary events, i.e. getting number 1, getting number 3 and getting number 5. (iii) Possible Event All the possible results, which may produced by the experiment are called possible events Example (i) On tossing a coin, we may get 2 possible results. H â€“ Getting Head (H) T â€“ Getting tail (T) So, number of possible events is 2, i.e. H and T. (ii) On throwing a dice, we may get 6 possible results. - 1 N Getting number 1 - 2 N Getting number 2 - 3 N Getting number 3 - 4 N Getting number 4 - 5 N Getting number 5 - 6 N Getting number 6. So, number of possible events is 6, i.e. 1 N , 2 N , 3 N , 4 N , 5 N , 6 N (iv) Favourable Events All the results, which may satisfy the given condition are called favourable events. Example (i) On tossing a coin and getting the tail, we have one favourable event. T = Getting tail (ii) On throwing a dice and getting a number greater than or equal to 5, we have two favourable events. - 5 N Getting number 5 - 6 N Getting number 6 3. Negation Of Event If E is an event, then negation of event E is NotE. In other words, the unfavourable results are called negation of event E. Negation of event is represented by a bar on the event. So, if event is E, then negation of event is E . Example On throwing a dice to get number 4. Event â€“ Getting number 4, i.e. 4 N So, Negation Of Event â€“ Getting a number other than 4, i.e. 2 1 N , N , 5 3 N , N and 6 N Probability Now, we will study Probability mathematically. That means, we will represent Probability in terms of numeric values. Let all the possible events = n Let all the favourable events = m Probability = P(A) Then, n m ) A ( P = So, if there are n possible events of a random experiment, out of which there are m favourable events, then probability is the ratio of possible events and favourable events. 1. If all possible events satisfy the condition, then the event is sure to happen. So, possible events = n Favourable events = m Then, probability P(A) = n m ? n n ) A ( P = [Possible Events = Favourable Event] ? 1 ) A ( P = ? Probability of an event sure to happen = 1 2. If no possible event satisfy the condition, then the event is sure not to happen. So, possible events = n Favourable event = m = 0 Then, probability n m ) A ( P = ? n 0 ) A ( P = [m = 0] ? 0 ) A ( P = ? Probability of an event sure not to happen =0 3. If there are n possible events and m favourable events, then there are m n - unfavourable events. So, possible events = n Favourable events = m Unfavourable events = n â€“ m Then, probability n m ) A ( P = â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (i) Also, probability of negation of event m m n ) A ( P - = â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (ii) So, n m n n m ) A ( P ) A ( P - + = + ? n m n m ) A ( P ) A ( P - + = + ? n n ) A ( P ) A ( P = + ? 1 ) A ( P ) A ( P = + That means, ) A ( P 1 ) A ( P - = Or ) A ( P 1 ) A ( P - = Example A card is drawn from a well-shuffled deck of playing cards. Find the probability of drawing: - (i) A face card. (ii) A red face card. We have, Total number of possible events ) n ( = Total Cards = 52 (i) Ace, king, queen and jack are called face cards. There are 4 aces, 4 kings, 4 queens, 4 jacks in a pack of cards. Then, Number of fvourable events (m) = Total Face Cards = 4 + 4 + 4 = 12 So, Probability of getting a face card n m ) A ( P = = 12 52 = 3 13 (ii) There are 2 red aces, 2 red kings, 2 red queens and 2 red jacks in a pack of cards. Then, Number of favourable events (m) =Total Number Of Red Faces = 2 + 2 + 2 + 2 =8 So, probability of getting a red face card n m ) A ( P = = 52 8 = 13 2Read More