UGC NET Questions (Paper – 1)

When two fair coins are tossed, the sample space consists of four equally likely outcomes: HH, HT, TH and TT. Exactly one head occurs in the outcomes HT and TH, giving 2 favourable cases. Therefore, the probability of exactly one head is 2/4, which simplifies to 1/2. Thus, 1/2 is the correct probability.

Option A:
A probability of 1/4 would correspond to just one favourable outcome out of four, but in this experiment there are two sequences that show exactly one head. Hence, 1/4 undercounts the favourable cases and is incorrect.

Option B:
1/2 is correct because the two favourable outcomes HT and TH out of four total outcomes give 2/4, which reduces to 1/2. This demonstrates a straightforward application of the probability formula for equally likely outcomes. Understanding such simple models helps in solving more advanced probability questions.

Option C:
3/4 would mean that three out of four outcomes have exactly one head, which is not true here. In reality, only two outcomes satisfy this condition, while HH and TT do not. Therefore, 3/4 exaggerates the likelihood and cannot be right.

Option D:
1/3 suggests an assumption of three equally likely outcomes, only one of which is favourable, which does not match the actual sample space of four outcomes. Thus, 1/3 is not compatible with the correct analysis of this experiment.

Inductive reasoning begins with particular observations and infers a general statement or rule from them. The conclusion goes beyond the information contained in the premises and is therefore only probable. This pattern is central to everyday learning and the scientific method. Hence reasoning from repeated specific cases to a general conclusion is called inductive reasoning.

Option A:
Option A, deductive, moves from general premises to specific conclusions, the reverse direction of what is described in the stem. Deductive arguments aim at necessity rather than probability. Thus deductive reasoning does not fit the reasoning pattern mentioned here.

Option B:
Option B, symbolic, refers to a formal way of representing arguments using symbols. It is a method of expression, not a distinct direction of reasoning from particular to general. Therefore symbolic reasoning is not the correct answer in this context.

Option C:
Option C, hypothetical, centres on assumptions and conditional statements but can be used in both deductive and inductive contexts. It does not specifically refer to drawing generalisations from specific instances. Hence hypothetical is not the best term for the reasoning described.

Option D:
Option D correctly identifies inductive reasoning, in which general laws are inferred from particular data. Such reasoning underlies empirical generalisations and statistical conclusions. Therefore inductive is the most appropriate choice for this question.

When a fair coin is tossed twice, the sample space has four equally likely outcomes: HH, HT, TH and TT. Exactly one head occurs in the outcomes HT and TH. There are therefore two favorable cases out of four possible outcomes. The required probability is the ratio of favorable outcomes to total outcomes, which is 2/4 = 1/2.

Option A:
Option A, 1/4, would represent the probability of a single specific outcome like HH or TT, not the event of exactly one head. It counts only one case instead of both HT and TH.

Option B:
Option B correctly identifies two favorable outcomes and divides by the four equally likely possibilities. This simplifies to 1/2, representing a fifty percent chance of getting exactly one head in two tosses.

Option C:
Option C, 3/4, would mean that three of the four outcomes give exactly one head, but HH and TT clearly do not fit that description. It overestimates the probability.

Option D:
Option D, 2/3, does not emerge from the simple fraction 2/4 and would incorrectly suggest that only three outcomes are being considered. It cannot be justified from the enumerated sample space.

A fair coin has two equally likely outcomes: head (H) and tail (T). The sample space thus contains two outcomes, and only one of them is a head. Probability is calculated as favourable outcomes divided by total outcomes, so P(H) = 1/2. Therefore, the probability of getting a head in a single toss is 1/2.

Option A:
Option A is correct because it directly follows from the definition of probability for equally likely events. With two outcomes and one favourable event, the fraction must be 1/2. This is one of the most basic probabilities and forms the foundation for more complex problems involving coins and other experiments.

Option B:
A probability of 1/3 would imply that there are three equally likely outcomes, only one of which is a head. However, a standard coin has only two sides, so this interpretation is impossible in this context. Hence, 1/3 cannot be correct.

Option C:
A probability of 1/4 suggests four equally likely outcomes, which might occur in other experiments but not with a single coin toss. Since there are only two outcomes here, 1/4 misrepresents the situation. Therefore, it is not the correct answer.

Option D:
A probability of 2/3 would mean that heads are more likely than tails, which contradicts the assumption of a fair coin with equal chances. The probability must be symmetric between head and tail, making 2/3 an invalid option.

When three fair coins are tossed, there are 2³ = 8 equally likely outcomes. The easiest way to find the probability of at least one head is to subtract the probability of getting no heads (i.e., all tails) from 1. The probability of all tails is 1 outcome (TTT) out of 8, which is 1/8. Therefore, the probability of at least one head is 1 − 1/8 = 7/8.

Option A:
Option A, 1/8, gives the probability of the complementary event (no head) rather than at least one head. It represents the chance of all tails.

Option B:
Option B, 3/8, might come from counting only single-head outcomes and ignoring cases with two or three heads, so it underestimates the event.

Option C:
Option C, 5/8, still misses some favourable outcomes and does not match the simple complement calculation.

Option D:
Option D correctly uses the complement rule and recognizes that all outcomes except the single all-tail result satisfy “at least one head,” leading to 7/8.

A standard deck of playing cards has 52 cards divided into 4 suits: hearts, diamonds, clubs and spades, each with 13 cards. The favorable outcomes for drawing a heart are 13. Thus, the probability is the ratio of favorable outcomes to total outcomes, which is 13/52. This fraction simplifies to 1/4. So, the probability of drawing a heart is 1/4.

Option A:
Option A, 1/4, is exactly the simplified fraction 13/52. It acknowledges that there are four equally sized suits and that hearts constitute one of the four, giving a 25% chance on a single draw.

Option B:
Option B, 1/3, would require three suits or some other distribution of cards, which is not the case in a standard deck.

Option C:
Option C, 1/13, incorrectly treats the number of ranks as the total sample space, ignoring that each suit has all 13 ranks.

Option D:
Option D, 3/13, would imply that hearts comprise nearly one-quarter of the cards, but with a different count than the actual 13 per suit. It is not supported by the structure of the deck.

Inductive reasoning starts from particular cases and moves toward general statements or laws. The conclusion goes beyond the information contained in the premises and is therefore only probable, not guaranteed. This pattern underlies many forms of empirical research and everyday generalisation. Thus the type of reasoning described in the stem is called inductive reasoning.

Option A:
Option A, deductive, moves in the opposite direction, from general premises to specific conclusions that are claimed to follow necessarily. Deduction does not characteristically build generalisations from particular observations. Hence deductive reasoning is not what the question refers to.

Option B:
Option B, analogical, relies on similarities between cases rather than directly aggregating instances into a general rule. While analogies can support general claims, analogical reasoning is defined by comparison rather than simple accumulation of observations. Therefore analogical is not the best answer here.

Option C:
Option C, causal, focuses on cause–effect relations and may be either deductive or inductive in form. Causal explanations explain why events occur but are not simply defined as moving from specific cases to general conclusions. Thus causal reasoning does not precisely match the stem.

Option D:
Option D is correct because inductive reasoning formally describes the process of inferring a general rule from specific examples. The scientific method often relies on such reasoning in developing hypotheses and theories from data.