Mdthbs

If we consider the first step (formation of carbocation by the attack of the proton).

A couple of textbooks I referred say that when the substituents on the $ce{C=C}$ are different, the direction of shift of the electron is decided by the inductive effect of the substituents. For example, in this image:

$$
begin{align}
ce{underset{1}{C}H3-underset{2}{C}H=underset{3}{C}H-underset{4}{C}H2-underset{5}{C}H3 &-> underset{1}{C}H3-underset{2}{overset{largeominus}{C}}H-underset{3}{overset{largeoplus}{C}}H-underset{4}{C}H2-underset{5}{C}H3} label{rxn:a}tag{a} \
ce{underset{1}{C}H3-underset{2}{C}H=underset{3}{C}H-underset{4}{C}H2-underset{5}{C}H3 &-> underset{1}{C}H3-underset{2}{overset{largeoplus}{C}}H-underset{3}{overset{largeominus}{C}}H-underset{4}{C}H2-underset{5}{C}H3} label{rxn:b}tag{b}
end{align}
$$

Assuming that a proton $ce{H+}$ is approaching, both the textbooks say that the electronic shift happens as in the case eqref{rxn:a} because eqref{rxn:a} has 2 ethyl groups to stabilize the carbocation by inductive effect. But isn't hyperconjugation more important than inductive effect in determining the stability?

By hyperconjugation, case eqref{rxn:b} should be the preferred pathway as there are 5 alpha-hydrogens (including the $ce{H}$ that is attacking and will join at the $ce{C-}$) and case eqref{rxn:a} has only 4? By this logic 2-bromopentane should dominate. Am I missing anything?

This sort of reaction isn't investigated seriously anymore. But 80 years ago, Kharasch et al. (J. Am. Chem. Soc. 1939, 61 (6), 1559–1564) wrote that

Both results agreed within the limit of error (3%) and indicated an equimolar mixture of 2- and 3-bromopentanes [...]

So, for all intents and purposes, there is no major product. It is not always possible to get a definitive answer based on simplified "rules" and concepts. All we can genuinely say in this case is that both intermediates are very similar in stability, and so both products will be formed in significant amounts. In fact, the carbocations may even interconvert rapidly by means of hydride shifts, because when pentan-2-ol or pentan-3-ol is treated with HBr, you get a mixture of the bromides (J. Am. Chem. Soc. 1930, 52 (6), 2440–2451).

This lack of selectivity is precisely why nobody investigates these seriously anymore. Maybe with modern analytical methods we could accurately measure it and find that the product ratio is 52:48, but what's the point? That's just as useless as a 50:50 mixture.

Your argument of inductive effect and hyperconjugation, made me to tell you a story about theory behind these two effects. I think after reading that you may able to understand how far we have come from them:

Wagner and Saytzeff (Ref.1; in German) prepared 2-pentene by dehydrohalogenation of 3-iodopentane. Addition of hydrogen iodide to this pentene yielded a product, which they characterized as the 2-iodopentane. From these results, they formulated the rule that the negative portion adds to the carbon atom bearing the shorter carbon chain.

Assuming that carbon compounds are polar in nature and using the work of Wagner and Saytzeff as experimental proof, Cuy (Ref.2) in 1920 proposed the hypothesis of alternatively charged carbon atoms. Accordingly, if hydrogen bromide (or other unsymmetrical reagent) is added to 2-pentene, the reaction would follow to give 2-bromopentane as the predominant product. It is a well-known fact (at the time as well) that primary propyl bromide in the presence of catalysts such as aluminum bromide, goes over to the secondary propyl bromide. Cuy had proposed that such isomeric rearrangements of alkyl halides can also be readily accounted for, on the basis of his hypothesis.

Lucas and Jameson (Ref.3) disagreed with Cuy’s theory of alternate polarization of carbon atoms, based on the fact that Wagner and Saytzeff’s reaction of hydrogen iodide and 2-pentene has given a mixture of isomeric iodides as products (Ref.1), not 2-iodopentane alone as Cuy assumed. Thus, they had advanced their own theory of electron displacement.

Lewis (Ref.4) was the first to show the effect of substituents upon the strength of organic acids, and to show that this effect extends throughout the entire carbon chain. Lucas and Jameson applied the work of Lewis to propene, to acrylic acid, and to dimethyl allene. They found that the addition of hydrogen halides could be explained by the theory of electronic displacement. The work of Lewis suggests that the alkyl group is more positive than hydrogen, while the carboxyl and halogen groups are more negative. Because of the positive character of the alkyl group as compared to hydrogen, the structural formula of propene can be written with the electrons constituting the double bond ($pi$-bond) being closer to the terminal carbon atom. Thus, addition of an unsymmetrical reagent would result in the more negative reagent attaching to the number two carbon. This theory also explains the rearrangement of 1-bromobutane to 2-bromobutane. Application of this theory to the addition of hydrogen bromide to 2-pentene would lead to a predominance of 3-bromopentane (Ref.5).

Lucas and Moyse (Ref.5) prepared the 2-pentene, and added hydrogen bromide to the olefin using glacial acetic acid as solvent. Using the refractive index method of analysis, they found 78% of the 3-bromo isomer and 22% of the 2-bromo isomer (74% overall yield). Lucas then considered the theory of electronic displacement confirmed (Lucas claimed the results obtained are in harmony with the hypothesis of electron displacement, but not with that of alternately polarized carbon atoms).

In l935, Baker and Nathan (Ref.6) advanced the theory of hyperconjugation in order to explain certain abnormal reactions in the halogenation of alkyl-substituted benzenes. They first postulated that the accelerating effects of alkyl groups must be related to their capacity for electron-release. The relative magnitude of such electron release by alkyl groups increases in the order $ce{C(CH3)3 < CH(CH3)2 < CH2CH3 < CH3}$. This order is exactly the reverse of that anticipated on the basis of the general inductive effect ($+I$) of an alkyl groups. They called this newfound electron releasing mechanism of the alkyl group attached to the necessary system is a type of tautomeric effect, which is often referred to as the Baker-Nathan effect in English literature for some extended time until they coined the word, hyperconjugation.

Further proof of this theory and beyond from a physical-chemical basis has been reviewed by Deasy (Ref.7) including quantum mechanics point of view. Basically, according to the theory of hyperconjugation, the addition of hydrogen bromide to 2-pentene would produce a predominane of 2-bromopentane, since it is possible to write three hyperoonjugative structures involving the three -hydrogens of the methyl group, but only two forms involving the two hydrogens of the ethyl group.

Based on above literature, the products outcome of this type of reaction is also depending on the conditions used, workup procedure, etc.

Now, it is well accepted that polarization can not be predicted, but product outcome can be determined by various factors, including hyperconjugation and 1,2-hydride shift (Ref.8).

References:

1. G. Wagner, A. Scytzeff, “Ueber amylenbromür und Amylglycol aus Diäthylcarbinol,” Justus Liebigs Annalen der Chemie 1875, 179(3), 302–313 (https://doi.org/10.1002/jlac.18751790305).


2. E. J. Cuy, “The electronic constitution of normal carbon chain compounds, saturated and unsaturated,” J. Am. Chem. Soc. 1920, 42(3), 503–514 (https://doi.org/10.1021/ja01448a016).


3. H. J. Lucas, A. Y. Jameson, “Electron Displacement in Carbon Compounds I. Electron Displacement Versus Alternate Polarity in Aliphatic Compounds,” J. Am. Chem. Soc. 1924, 46(11), 2475–2482 (https://doi.org/10.1021/ja01676a018).


4. G. N. Lewis, “The Atom and the Molecule,” J. Am. Chem. Soc. 1916, 38(4), 762–785 (https://doi.org/10.1021/ja02261a002).


5. H. J. Lucas, H. W. Moyse, “Electron Displacement in Carbon Compounds II. Hydrogen Bromide and 2-Pentene,” J. Am. Chem. Soc. 1925, 47(5), 1459–1461 (https://doi.org/10.1021/ja01682a037).


6. J. W. Baker, W. S. Nathan, “429. The mechanism of aromatic side-chain reactions with special reference to the polar effects of substituents. Part V. The polar effects of alkyl groups,” J. Chem. Soc. 1935, 1844 –1847 (DOI:10.1039/JR9350001844).


7. C. L. Deasy, “Hyperconjugation,” Chem. Rev. 1945, 36(2), 145 –155 (https://doi.org/10.1021/cr60114a001).


8. B. A. Howell, R. E. Kohrman, “Preparation of 2-bromopentane,” J. Chem. Educ. 1984, 61(10), 932–934 (https://doi.org/10.1021/ed061p932).

Well the thing here is that you mix up thermodynamic data (one is more stable) to kinetic data (a proton is approaching).

They are two different processes.