Heads and Runs in Multiple Tossing of a Coin and Contingency Table, Statistical Note 40

List the possible outcomes in three tosses of a coin, categorize them by the number of number of heads and the number of runs in the sequence and prepare a contingency table.

Tossing of a coin thrice has two to the power three equal to eight possible outcomes (refer to my statistical note 9). Every outcome is measured using two discrete random variables – the number of heads and the number of runs. The outcomes have number of heads ranging from zero to three as mutually exclusive categories of response as shown in Table 1. A run is a sequence of flips of the same face of a coin. The number of runs ranges from one to three as mutually exclusive categories of response (Table 1). Example, an outcome HTH has three runs, as every toss has a different face than the previous toss. A contingency table, also known as the cross tabulation, crosstab or two-way table counts the number observations for each category of two variables.

Table 1: Possible outcomes in three tosses of a coin by number of heads and number of runs

The contingency table presents the number of possible outcomes by number of heads and number of runs (Table 2).

Central values of both variables have high probability of occurrence. Example, outcomes with one and two heads respectively have higher probability of occurrence, three by eight. Likewise, outcomes with two runs have higher probability of occurrence, four by eight. Refer to my statistical note 3 for probability and contingency table.

Multiplication Rule of Dependent Events, Statistical Note 39

If two cards are drawn without replacement from a deck of cards, what is the joint probability that both cards are red?

The multiplication rule of probability of two dependent events is the product of the probability of first independent marginal event and the conditional probability of second dependent marginal event. Same applies to the joint probability of multiple dependent events.

Symbolically, let A be an independent event and B be a dependent event, the joint probability that both events and A and B occur is stated as:

P(A and B) = P(A)*P (B/A), where P(A) is the probability of an independent event A and P(B/A) is the conditional probability of a dependent event B given the occurrence of an independent event A.

The events are said to be dependent events if the occurrence of the first event affects the probability of the second and following events. In this example, drawing of a red card in the first event and not replacing back that card to the deck of cards before second draw of a card affects the second draw of a card. This applies to multiple draws of cards without replacement from a deck of cards.

Let R1 be an independent event of drawing a red card in the first draw of a card.  There are 26 red cards in a deck of cards. The marginal probability of R1, P(R1) is 26 divided by 52, 0.5. Let R2 be a dependent event of drawing a red card from the deck of cards in which the first drawn card is not replaced back.  There are 25 red cards left in the deck of 51 cards, as the first drawn cards is red which is not replaced back in the deck. Thus, the conditional probability of R2 given R1 in the first draw, P(R2/R1) is 25 divided by 51, 0.4902.

Now, applying the multiplication rule of dependent events, the joint probability that red cards are drawn in both draws without replacement is the product of P(R1) in the first draw and the P(R2) in the second draw.

Symbolically, P(R1 and R2) = P(R1)*P (R2/R1)=26/52 *25/51 =0.2450

The total number of outcomes in two draws of cards without replacement from a deck of cards and their probabilities are shown using a tree diagram below:

Figure 1: Marginal probabilities of first draws and conditional probabilities of second draws given the first draws of cards without replacement and joint probabilities of possible outcomes

Time to Break Gender Stereotype of All Kinds

There could be gender stereotypes in some professions in the past leading people to perceive a man is a doctor and a woman a nurse, a man is a pilot and a woman an airhostess. However, there is nothing that men can do and women cannot or vice versa in this millennium. The constitution clearly prohibits against discriminating on the grounds of gender in remuneration and social security for the same work. A male doctor can be a gynecologist and a female doctor can treat a male reproductive organ.  Equal opportunity for men and women needs to be given in every profession. Thus, gender stereotypes need to be discouraged by creating awareness, creating opportunities and taking legal actions against the violation of the rules.

Basan Shrestha, Ghattekulo, Kathmandu

The Himalayan Times, PeopleSpeak, May 12, 2019

http://epaper.thehimalayantimes.com/html5/reader/production/default.aspx?pubname=&pubid=cd7278e2-4150-475f-8abe-305e5ed57783

Multiplication Rule of Independent Events, Statistical Note 38

If an unbiased coin with two sides (Head and Tail) is tossed twice, what is the joint probability that head appears in both tosses?

The multiplication rule of probability of two independent events is the product of the probability of first independent marginal event and the probability of second independent marginal event. Same applies to the joint probability of multiple independent events.

Symbolically, if two events A and B are independent, the probability that both events and A and B occur is stated as:

P(A and B) = P(A)*P (B)

The events are said to be independent events if the occurrence of the first event does not affect the probability of the second and following events. In this example, occurrence of head or tail in the first toss does not affect the occurrence of head or tail in the second toss. Any of head or tail could appear in any number of toss of a coin. This applies to multiple tosses of a coin.

Let H be an independent event of turning head in a toss of a coin.  Likewise, let T be an independent event of turning tail in a toss of a coin.  The marginal probability of H, P(H) is one by two, 0.5. Similarly, the marginal probability of T, P(T) is one by two, 0.5.

The total number of outcomes in two tosses of a coin can be shown a tree diagram like the one shown below:

Figure 1: Marginal probabilities of independent events in first and second tosses of a coin and number of outcomes

Now, applying the multiplication rule of independent events, the joint probability that head appears in both tosses is the product of P(H) in the first toss and the P(H) in the second toss.

Symbolically, P(H and H) = ½ * ½ = ¼ = 0.25

Alternatively, the occurrence of head in both tosses is Outcome 1 of four outcomes in the tree diagram, and thus the probability is one out of four, equal to 0.25.